MGU Mathematics Series — Algebra II
Part IV: Exponential, Logarithmic, and Sequence Models
Logarithms · Properties · Equations · Logarithmic Scales
Chapter 10 of 14
MGU Mathematics Series  ·  Algebra II  ·  Library Textbook Edition

Chapter 10:
Logarithmic
Functions

The inverse of exponentiation — evaluating, graphing, applying the three core properties, solving exponential and logarithmic equations, and interpreting logarithmic scales

■ Math Level: Algebra II ■

■ 14 Sections  •  Part IV of 5 ■

Follows Chapter 9: Exponential Functions  ·  Prepares students for Chapter 11: Sequences and Series

Chapter 10 — Front Matter

Purpose and Overview

Chapter 9 introduced exponential functions as models of multiplicative change. You learned that exponential functions grow or decay at a constant percent rate, and that they appear in compound interest, population growth, radioactive decay, and dozens of other real situations. You also saw that exponential equations can be difficult to solve when the variable is in the exponent. Asking a question like "How long does it take for a population to triple?" or "When will a medication concentration drop to a safe level?" requires a mathematical tool that can isolate an exponent.

That tool is the logarithm.

Chapter 10 introduces logarithms as the inverse operation of exponentiation. A logarithm answers the question: what exponent was used? This simple idea connects to graphing, equation solving, measurement, science, technology, finance, and any field where quantities change exponentially. Students who understand logarithms gain access to one of the most practically powerful ideas in all of mathematics.

This chapter will teach you to evaluate logarithms, recognize common and natural logarithms, graph logarithmic functions, apply the three core logarithm properties, solve exponential and logarithmic equations, and interpret logarithmic scales. Every idea in this chapter grows directly from the definition: a logarithm is an exponent.

Opening Question

A certain bacteria colony doubles in size every three hours. You start with 500 bacteria. A researcher wants to know: how many hours will it take for the population to reach 64,000?

You could write the equation \(500 \cdot 2^t = 64{,}000\). Dividing both sides by 500 gives \(2^t = 128\). Now you need to find the value of \(t\) such that 2 raised to the power \(t\) equals 128. You might recognize that \(2^7 = 128\), so \(t = 7\), meaning 21 hours. But what if the target population were 100,000, or 1,000,000? What if the growth rate were not as clean as doubling? You would need a systematic way to find an exponent — and that is exactly what logarithms provide.

Why This Chapter Matters. Logarithms are not a mathematical curiosity. They are one of the primary tools used in science, engineering, finance, music, acoustics, seismology, medicine, and computer science. The Richter scale for earthquakes, the decibel scale for sound, the pH scale for chemistry, and the magnitude scale for stars are all logarithmic. Financial analysts use logarithms to model continuous growth. Computer scientists use logarithms to describe algorithmic efficiency. Engineers use logarithms to analyze signals and systems.

Beyond applications, logarithms complete your understanding of inverse functions. In Algebra I, you learned that square roots and squares are inverses, and that addition and subtraction undo each other. Logarithms and exponential functions are inverses in exactly the same sense. Understanding this relationship gives you a full picture of both function families, which you will continue to use in Precalculus, Calculus, Statistics, and Physics.

14Sections
10Objectives
17Key Terms
14+Worked Examples

Learning Objectives

  1. 1Explain what a logarithm represents and convert fluently between logarithmic and exponential notation.
  2. 2Evaluate basic logarithmic expressions without a calculator.
  3. 3Distinguish between common logarithms and natural logarithms; apply the Change of Base Formula.
  4. 4Graph logarithmic functions and describe their key features: domain, range, asymptote, and intercept.
  5. 5Apply transformations to logarithmic functions and identify the effect on the graph.
  6. 6Use the Product, Quotient, and Power Properties of Logarithms to expand and condense expressions.
  7. 7Solve exponential equations using logarithms.
  8. 8Solve logarithmic equations and check for extraneous solutions.
  9. 9Interpret and apply logarithmic scales in real-world contexts.
  10. 10Identify and avoid the most common logarithm mistakes.
Chapter 10 — Contents

Sections at a Glance

Sections 10.1 – 10.5
The Logarithm: Definition, Notation, and Evaluation
Why Logarithms · Definition · Common Log · Natural Log · Evaluating
Section 10.1

Why Logarithms Are Needed

You have already solved equations such as \(3x + 5 = 20\) or \(x^2 = 49\). The first uses subtraction and division to isolate \(x\). The second uses a square root, which is the inverse of squaring. Both strategies rely on an inverse operation — a procedure that undoes what was done to the variable.

Now consider the equation \(2^x = 50\). The variable \(x\) is in the exponent. The operations you know from equation solving — adding, subtracting, multiplying, dividing, taking roots — do not directly isolate an exponent. You need an inverse operation for exponentiation. That inverse is the logarithm.

Recall that an exponential function has the form \(f(x) = b^x\), where \(b > 0\) and \(b \neq 1\). The input \(x\) is the exponent; the output is the result of raising \(b\) to that power. To undo this, you need a function that takes the output and returns the exponent. That function is the logarithm.

In plain language: the logarithm asks — what power do I raise the base to in order to get this number?

A Simple Illustration

Since \(3^4 = 81\), the reverse question is: what power of 3 equals 81? The answer is 4.

The logarithm records that answer: the logarithm base 3 of 81 is 4.

Similarly: \(10^3 = 1000\) → the log base 10 of 1000 is 3.    \(2^5 = 32\) → the log base 2 of 32 is 5.

The pattern: you always know the base and the result, and the logarithm returns the exponent.

Why This Matters Beyond Algebra

How long does it take for a drug to decay to half its dose? How many times must you double a population before it exceeds a threshold? How far apart in magnitude are two earthquakes? Logarithms convert these exponent questions into solvable arithmetic. They unlock the structure inside exponential relationships.

Practice 10.1Concept Check
  1. Explain in your own words why you cannot solve \(2^x = 50\) using the same inverse operations used to solve \(2x = 50\).
  2. What question does a logarithm answer? Write your answer as a complete sentence.
  3. If \(5^3 = 125\), what reverse question does the logarithm base 5 of 125 answer? What is its value?
Section 10.2

Logarithms as Exponents

Formal Definition

Let \(b > 0\), \(b \neq 1\), and \(x > 0\). The logarithm base \(b\) of \(x\) is the exponent to which \(b\) must be raised to produce \(x\).

\(\log_b(x) = y \quad \text{means} \quad b^y = x\)

Read \(\log_b(x)\) as "log base \(b\) of \(x\)." The number \(b\) is the base. The number \(x\) is the argument. The number \(y\) is the logarithm — the exponent.

The Two Forms

Every logarithmic statement has a corresponding exponential statement, and vice versa. Converting fluently between these two forms is the most fundamental skill in this chapter.

Exponential FormLogarithmic Form
\(b^y = x\)\(\log_b(x) = y\)
\(2^3 = 8\)\(\log_2(8) = 3\)
\(10^2 = 100\)\(\log_{10}(100) = 2\)
\(5^0 = 1\)\(\log_5(1) = 0\)
\(3^{-1} = \tfrac{1}{3}\)\(\log_3\!\left(\tfrac{1}{3}\right) = -1\)
Example 10.2.1
Converting exponential to logarithmic form
  • ProblemWrite \(4^3 = 64\) in logarithmic form.
  • ThinkIdentify the base (4), the exponent (3), and the result (64). The logarithm asks: what power of 4 gives 64?
  • Solve\(4^3 = 64 \quad\longrightarrow\quad \log_4(64) = 3\)
  • CheckDoes \(4^3 = 64\)? Yes. ✓
Answer: \(\log_4(64) = 3\)
Example 10.2.2
Converting logarithmic to exponential form
  • ProblemWrite \(\log_2(32) = 5\) in exponential form.
  • ThinkThe base is 2, the logarithm (exponent) is 5, and the argument is 32. The exponential form raises the base to the exponent.
  • Solve\(\log_2(32) = 5 \quad\longrightarrow\quad 2^5 = 32\)
  • Check\(2^5 = 32\) ✓
Answer: \(2^5 = 32\)
Two Useful Special Cases

\(\log_b(1) = 0 \quad\) because \(\quad b^0 = 1\) for any valid base.

\(\log_b(b) = 1 \quad\) because \(\quad b^1 = b\) for any valid base.

These are not rules to memorize in isolation — they are consequences of the definition.

Domain Restriction

The argument of a logarithm must be a positive number. \(\log_b(0)\) is undefined — no exponent raises \(b\) to 0. \(\log_b(-5)\) is undefined — no real exponent produces a negative output when the base is positive.

Practice 10.2Converting Between Forms

Write each in the other form (exponential ↔ logarithmic).

  1. \(5^3 = 125\)
  2. \(2^{-3} = \frac{1}{8}\)
  3. \(10^4 = 10{,}000\)
  4. \(e^1 = e\)
  5. \(\log_6(36) = 2\)
  6. \(\log_3(81) = 4\)
  7. \(\log_{10}(0.01) = -2\)
  8. \(\log_7(1) = 0\)
Section 10.3

Common Logarithms

In everyday mathematics, science, and engineering, the most frequently used base is 10. The common logarithm is the logarithm base 10. Because it is so common, mathematicians adopted a shorthand: when no base is written, the base is assumed to be 10.

\[\log(x) = \log_{10}(x)\]

This convention appears throughout textbooks, calculators, scientific instruments, and real-world scales. When you see \(\log(x)\) without a subscript base, always read it as the base-10 logarithm.

ExpressionValueReasoning
\(\log(1)\)0\(10^0 = 1\)
\(\log(10)\)1\(10^1 = 10\)
\(\log(100)\)2\(10^2 = 100\)
\(\log(1{,}000)\)3\(10^3 = 1{,}000\)
\(\log(0.1)\)−1\(10^{-1} = 0.1\)
\(\log(0.01)\)−2\(10^{-2} = 0.01\)
Example 10.3.1
Evaluating common logarithms
  • ProblemEvaluate: (a) \(\log(10{,}000)\)   (b) \(\log(0.001)\)   (c) \(\log(1)\)
  • ThinkFor each, ask: what power of 10 equals the argument?
  • Solve (a) \(\log(10{,}000) = \log(10^4) = 4\) (b) \(\log(0.001) = \log(10^{-3}) = -3\) (c) \(\log(1) = \log(10^0) = 0\)
  • Check\(10^4 = 10{,}000\) ✓   \(10^{-3} = 0.001\) ✓   \(10^0 = 1\) ✓
Answer: (a) 4   (b) −3   (c) 0
Order of Magnitude

The common logarithm is closely related to order of magnitude — how large a number is in terms of powers of 10. Numbers like 1, 10, 100, 1000 differ by one order of magnitude each, which corresponds to a difference of 1 in their common logarithms. This is why logarithms appear on scales where quantities span many orders of magnitude.

Practice 10.3Common Logarithms
  1. \(\log(100{,}000)\)
  2. \(\log(0.0001)\)
  3. \(\log(10^7)\)
  4. \(\log(10^{-4})\)
  5. \(\log(1)\)
  6. \(\log(10)\)
Section 10.4

Natural Logarithms

In Chapter 9, you encountered the number \(e \approx 2.71828\), the base of continuous growth. Because \(e\) is so fundamental, mathematicians gave a special name to the logarithm with base \(e\).

The natural logarithm is the logarithm base \(e\). It is written \(\ln(x)\), read as "the natural log of \(x\)."

\[\ln(x) = \log_e(x)\]

Everything you know about logarithms applies to \(\ln(x)\). The only difference is the base.

\[\ln(x) = y \quad\text{means}\quad e^y = x\]

The special cases follow immediately: \(\ln(1) = 0\) because \(e^0 = 1\); and \(\ln(e) = 1\) because \(e^1 = e\); and \(\ln(e^k) = k\) for any real \(k\).

ExpressionValueReasoning
\(\ln(1)\)0\(e^0 = 1\)
\(\ln(e)\)1\(e^1 = e\)
\(\ln(e^3)\)3\(e^3 = e^3\)
\(\ln(e^{-2})\)−2\(e^{-2} = e^{-2}\)
\(\ln(10)\)≈ 2.303Calculator
\(\ln(2)\)≈ 0.693Calculator
Example 10.4.1
Evaluating natural logarithms
  • ProblemEvaluate: (a) \(\ln(e^5)\)   (b) \(\ln(1)\)   (c) \(\ln\!\left(\dfrac{1}{e^2}\right)\)
  • ThinkUse the inverse relationship between \(\ln\) and \(e^x\). For (c), rewrite the fraction using a negative exponent.
  • Solve (a) \(\ln(e^5) = 5\) (b) \(\ln(1) = 0\) (c) \(\ln\!\left(\dfrac{1}{e^2}\right) = \ln(e^{-2}) = -2\)
  • Check\(e^5 = e^5\) ✓   \(e^0 = 1\) ✓   \(e^{-2} = \frac{1}{e^2}\) ✓
Answer: (a) 5   (b) 0   (c) −2
When to Use log vs. ln

Use \(\log\) (base 10) when working with measurement scales, powers of 10, or common scientific applications. Use \(\ln\) (base \(e\)) when working with continuous growth, decay, and most calculus applications. In this chapter, you will work with both.

MGU Library Connection
  • The number \(e\) and the natural logarithm reappear extensively in Calculus I, where derivatives and integrals of exponential and logarithmic functions are central topics
  • The natural logarithm has a particularly elegant calculus identity that makes it essential throughout all higher mathematics
  • MGU Physics and Engineering courses use \(\ln\) in decay models, signal analysis, and thermodynamics
Practice 10.4Natural Logarithms
  1. \(\ln(e^7)\)
  2. \(\ln(e^{-1})\)
  3. \(\ln(e^{1/2})\)
  4. \(\ln(1)\)
  5. \(\ln(e^0)\)
  6. \(\ln\!\left(\dfrac{1}{e^4}\right)\)
Section 10.5

Evaluating Logarithms

Strategy: Converting to Exponential Form

To evaluate \(\log_b(x)\):

1. Set the expression equal to \(y\): \(\log_b(x) = y\)

2. Convert to exponential form: \(b^y = x\)

3. Ask: what power of \(b\) gives \(x\)?

4. Write the value of \(y\).

Example 10.5.1
Evaluating a logarithm
  • ProblemEvaluate \(\log_3(81)\).
  • ThinkAsk: what power of 3 equals 81?
  • Solve Let \(\log_3(81) = y\). Then \(3^y = 81\). Since \(3^4 = 81\), \quad y = 4\).
  • Check\(3^4 = 81\) ✓
Answer: \(\log_3(81) = 4\)
Example 10.5.2
Evaluating a logarithm with a fraction
  • ProblemEvaluate \(\log_5\!\left(\dfrac{1}{125}\right)\).
  • ThinkWhat power of 5 gives \(\frac{1}{125}\)? Since \(5^3 = 125\), think about negative exponents.
  • Solve Let \(\log_5\!\left(\dfrac{1}{125}\right) = y\). Then \(5^y = \dfrac{1}{125}\). Since \(\dfrac{1}{125} = 5^{-3}\), \quad y = -3\).
  • Check\(5^{-3} = \dfrac{1}{5^3} = \dfrac{1}{125}\) ✓
Answer: \(\log_5\!\left(\dfrac{1}{125}\right) = -3\)
Example 10.5.3
Evaluating when both base and argument share a common base
  • ProblemEvaluate \(\log_4(8)\).
  • Think8 does not look like a power of 4, but both 4 and 8 are powers of 2. Rewrite both using base 2.
  • Solve \(4 = 2^2\) and \(8 = 2^3\). Let \(\log_4(8) = y\). Then \(4^y = 8\), so \((2^2)^y = 2^3\), giving \(2^{2y} = 2^3\). Since bases match: \(2y = 3\), so \(y = \dfrac{3}{2}\).
  • Check\(4^{3/2} = (\sqrt{4})^3 = 2^3 = 8\) ✓
Answer: \(\log_4(8) = \dfrac{3}{2}\)

Change of Base Formula

When you need to evaluate a logarithm with a base that is not 10 or \(e\), and the argument is not an exact power of the base, convert using the Change of Base Formula:

Change of Base Formula

\[\log_b(x) = \frac{\log(x)}{\log(b)} = \frac{\ln(x)}{\ln(b)}\]

You may use either common or natural logarithms — both give the same result. This allows any logarithm to be evaluated with a standard calculator.

Example 10.5.4
Using the Change of Base Formula
  • ProblemEvaluate \(\log_7(200)\). Round to three decimal places.
  • Think200 is not a simple power of 7, so use the Change of Base Formula with common logarithms.
  • Solve\[\log_7(200) = \frac{\log(200)}{\log(7)} = \frac{2.30103}{0.84510} \approx 2.724\]
  • CheckDoes \(7^{2.724} \approx 200\)? Using a calculator: yes. ✓
Answer: \(\log_7(200) \approx 2.724\)
Practice 10.5Evaluating Logarithms

Evaluate each without a calculator. Show the exponential form.

  1. \(\log_2(16)\)
  2. \(\log_3(27)\)
  3. \(\log_5(125)\)
  4. \(\log_4(64)\)
  5. \(\log_7(1)\)
  6. \(\log_6(6)\)
  7. \(\log_2\!\left(\tfrac{1}{8}\right)\)
  8. \(\log_9(3)\)
  9. \(\log_{1/2}(16)\)
  10. \(\log(10^{-5})\)

Use the Change of Base Formula. Round to four decimal places.

  1. \(\log_3(20)\)
  2. \(\log_7(100)\)
  3. \(\log_6(1{,}000)\)
  4. \(\log_5(80)\)
Sections 10.6 – 10.7
Graphing and Transforming Logarithmic Functions
Graph Shape · Key Features · Asymptotes · Transformations
Section 10.6

Graphing Logarithmic Functions

Because the logarithmic function is the inverse of the exponential function, its graph is the reflection of the exponential graph across the line \(y = x\). The parent function of the exponential family is \(f(x) = b^x\). Its inverse is \(f^{-1}(x) = \log_b(x)\).

Key Features of \(y = \log_b(x)\) when \(b > 1\)

Domain: \(x > 0\) — only positive inputs are defined

Range: all real numbers

x-intercept: \((1,\,0)\) — because \(\log_b(1) = 0\)

Vertical asymptote: \(x = 0\) (the y-axis) — the function approaches but never reaches \(x = 0\) from the right

Behavior: increasing for all \(x > 0\), but more and more slowly as \(x\) grows

No y-intercept: the domain does not include \(x = 0\)

Diagram Instruction

Draw a coordinate plane with two curves. One curve is \(y = \log_2(x)\), which passes through \((1, 0)\), \((2, 1)\), \((4, 2)\), and \((8, 3)\), rising steeply near the y-axis then flattening. The other curve is the reflection across the line \(y = x\), which is the exponential function \(y = 2^x\). Label the vertical asymptote of the logarithm as \(x = 0\) and the horizontal asymptote of the exponential as \(y = 0\). Mark the point \((1, 0)\) on the logarithmic curve and \((0, 1)\) on the exponential curve, illustrating the reflection.

Example 10.6.1
Graphing \(y = \log_3(x)\)
  • ProblemGraph \(y = \log_3(x)\). Identify the domain, range, x-intercept, and vertical asymptote.
  • ThinkGenerate a table using \(3^y = x\). Choose integer values of \(y\) to get clean values of \(x\).
  • Solve \(y = -2 \Rightarrow x = \tfrac{1}{9}\);   \(y = -1 \Rightarrow x = \tfrac{1}{3}\);   \(y = 0 \Rightarrow x = 1\);   \(y = 1 \Rightarrow x = 3\);   \(y = 2 \Rightarrow x = 9\);   \(y = 3 \Rightarrow x = 27\) Plot the points \(\bigl(\tfrac{1}{9},\,-2\bigr)\), \(\bigl(\tfrac{1}{3},\,-1\bigr)\), \((1,\,0)\), \((3,\,1)\), \((9,\,2)\), \((27,\,3)\). Draw a smooth curve approaching the y-axis without touching it.
  • AnswerDomain: \((0,\,\infty)\). Range: all real numbers. x-intercept: \((1,\,0)\). Vertical asymptote: \(x = 0\).
The graph is an increasing curve defined for \(x > 0\), crossing the x-axis at \((1,\,0)\), with a vertical asymptote at \(x = 0\).
Practice 10.6Graphing Logarithmic Functions
  1. Graph \(y = \log_2(x)\). Create a table with at least five points. Identify the domain, range, vertical asymptote, and x-intercept.
  2. Graph \(y = \log(x)\) and \(y = \ln(x)\) on the same coordinate plane. Describe the similarities and differences.
  3. Describe the end behavior of \(y = \log_b(x)\) as \(x \to 0^+\) and as \(x \to \infty\). How does this compare to \(y = b^x\)?
  4. Without graphing, state the domain and x-intercept of \(y = \log_5(x)\). Explain your reasoning.
Section 10.7

Logarithmic Transformations

Logarithmic functions follow the same transformation rules as all functions you have studied. Starting from the parent function \(y = \log_b(x)\), each modification to the equation shifts, stretches, compresses, or reflects the graph. The general transformed form is:

\[y = a \cdot \log_b(x - h) + k\]

ParameterEffect on GraphEffect on Asymptote / Domain
\(h > 0\)Shift right \(h\) unitsAsymptote moves to \(x = h\); domain \((h,\infty)\)
\(h < 0\)Shift left \(|h|\) unitsAsymptote moves to \(x = h\)
\(k > 0\)Shift up \(k\) unitsNo effect on asymptote or domain
\(k < 0\)Shift down \(|k|\) unitsNo effect on asymptote or domain
\(a > 1\)Vertical stretchNo effect on asymptote or domain
\(a < 0\)Reflection over the x-axisNo effect on asymptote or domain
Example 10.7.1
Graphing a transformed logarithmic function
  • ProblemGraph \(y = \log_2(x - 3) + 1\). Identify the vertical asymptote, domain, and x-intercept.
  • ThinkStart from \(y = \log_2(x)\). The \(-3\) inside shifts the graph right 3; the \(+1\) outside shifts up 1. The vertical asymptote moves from \(x = 0\) to \(x = 3\). The domain becomes \(x > 3\).
  • Solve Apply the shifts to parent function points: \((1,0) \to (4,1)\); \((2,1) \to (5,2)\); \((4,2) \to (7,3)\); \((\frac{1}{2},-1) \to (3.5,0)\). Find the x-intercept by setting \(y = 0\): \[0 = \log_2(x-3)+1 \;\Rightarrow\; -1 = \log_2(x-3) \;\Rightarrow\; 2^{-1} = x-3 \;\Rightarrow\; x = 3.5\]
Vertical asymptote: \(x = 3\). Domain: \((3, \infty)\). x-intercept: \((3.5,\,0)\).
Diagram Instruction

Draw a coordinate plane. Draw the transformed curve passing through \((3.5,\,0)\), \((4,\,1)\), \((5,\,2)\), and \((7,\,3)\). Draw a dashed vertical line at \(x = 3\) as the asymptote. The curve should approach this line from the right without crossing it. Label the asymptote and the x-intercept.

Finding the Domain of a Transformed Logarithmic Function

To find the domain, identify the vertical asymptote \(x = h\). For \(b > 1\), the domain is always to the right of that asymptote: \(x > h\). Always set the argument of the logarithm greater than zero and solve — that inequality defines the domain.

Practice 10.7Transformations

For each function, identify the transformation from the parent, state the vertical asymptote, domain, and x-intercept.

  1. \(y = \log_2(x) - 3\)
  2. \(y = \log_2(x + 4)\)
  3. \(y = -\log_3(x)\)
  4. \(y = 2\log(x-1) + 3\)
  5. \(y = \log_5(x + 2) - 1\)
  6. \(y = \ln(x - 4) + 2\)
Sections 10.8 – 10.10
The Three Properties of Logarithms
Product Property · Quotient Property · Power Property
Section 10.8

Product Property of Logarithms

The three logarithm properties — Product, Quotient, and Power — are derived from the corresponding exponent rules. Understanding where they come from makes them far easier to remember and apply correctly.

The exponent rule for products is \(b^m \cdot b^n = b^{m+n}\). Because logarithms are exponents, this multiplication of the base becomes addition of the logarithms.

Product Property

For any valid base \(b\) and any positive numbers \(M\) and \(N\):

\[\log_b(MN) = \log_b(M) + \log_b(N)\]

The logarithm of a product equals the sum of the logarithms.

Example 10.8.1
Expanding with the Product Property
  • ProblemExpand \(\log_4(16 \cdot 64)\).
  • ThinkApply the Product Property to split the product into a sum.
  • Solve \(\log_4(16 \cdot 64) = \log_4(16) + \log_4(64) = 2 + 3 = 5\) Because \(4^2 = 16\) and \(4^3 = 64\).
  • Check\(16 \cdot 64 = 1024 = 4^5\), so \(\log_4(1024) = 5\) ✓
Answer: 5
Example 10.8.2
Expanding logarithms with variables
  • ProblemExpand \(\ln(5x^3)\).
  • ThinkUse the Product Property to separate the 5 and \(x^3\). Then use the Power Property on the exponent.
  • Solve\(\ln(5x^3) = \ln(5) + \ln(x^3) = \ln(5) + 3\ln(x)\)
Answer: \(\ln(5) + 3\ln(x)\)
Student Warning

The Product Property applies to the logarithm of a product, not to the product of two separate logarithms. \(\log_b(M) \cdot \log_b(N)\) has no standard simplification and does not equal \(\log_b(MN)\).

Section 10.9

Quotient Property of Logarithms

The exponent rule for quotients is \(\dfrac{b^m}{b^n} = b^{m-n}\). Division of outputs corresponds to subtraction of exponents — and the same structure appears in logarithms.

Quotient Property

For any valid base \(b\) and any positive numbers \(M\) and \(N\):

\[\log_b\!\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)\]

The logarithm of a quotient equals the difference of the logarithms.

Example 10.9.1
Expanding with the Quotient Property
  • ProblemExpand \(\log\!\left(\dfrac{100}{x}\right)\).
  • Solve\[\log\!\left(\frac{100}{x}\right) = \log(100) - \log(x) = 2 - \log(x)\]
Answer: \(2 - \log(x)\)
Example 10.9.2
Condensing with the Quotient Property
  • ProblemWrite \(\ln(x) - \ln(3)\) as a single logarithm.
  • ThinkA difference of logarithms with the same base can be written as the logarithm of a quotient.
  • Solve\(\ln(x) - \ln(3) = \ln\!\left(\dfrac{x}{3}\right)\)
Answer: \(\ln\!\left(\dfrac{x}{3}\right)\)
Student Warning — Subtraction Inside vs. Outside

Be careful: \(\log_b(M) - \log_b(N) = \log_b\!\left(\dfrac{M}{N}\right)\). The subtraction is outside the logarithm in expanded form and becomes division inside the argument when condensed. This is not the same as \(\log_b(M - N)\).

Section 10.10

Power Property of Logarithms

The exponent rule for powers is \((b^m)^n = b^{mn}\). When a power appears inside a logarithm, the exponent can move outside as a multiplier.

Power Property

For any valid base \(b\), any positive number \(M\), and any real number \(n\):

\[\log_b(M^n) = n \cdot \log_b(M)\]

An exponent inside the argument of a logarithm can be moved to the front as a coefficient.

Example 10.10.1
Using the Power Property
  • ProblemWrite \(\log_2(x^5)\) in expanded form.
  • Solve\(\log_2(x^5) = 5 \cdot \log_2(x)\)
Answer: \(5\log_2(x)\)
Example 10.10.2
Expanding a complex logarithm using all three properties
  • ProblemExpand \(\log\!\left(\dfrac{x^3 y^2}{z}\right)\) completely.
  • ThinkUse the Quotient Property first, then the Product Property on the numerator, then the Power Property on each power.
  • Solve \[\log\!\left(\frac{x^3 y^2}{z}\right) = \log(x^3 y^2) - \log(z)\] \[= \log(x^3) + \log(y^2) - \log(z)\] \[= 3\log(x) + 2\log(y) - \log(z)\]
Answer: \(3\log(x) + 2\log(y) - \log(z)\)
Example 10.10.3
Condensing logarithms into a single expression
  • ProblemWrite \(3\ln(x) - 2\ln(y) + \ln(5)\) as a single logarithm.
  • ThinkUse the Power Property in reverse to move coefficients to exponents, then use the Product and Quotient Properties to combine.
  • Solve \[3\ln(x) - 2\ln(y) + \ln(5) = \ln(x^3) - \ln(y^2) + \ln(5)\] \[= \ln\!\left(\frac{x^3}{y^2}\right) + \ln(5) = \ln\!\left(\frac{5x^3}{y^2}\right)\]
Answer: \(\ln\!\left(\dfrac{5x^3}{y^2}\right)\)
Summary of the Three Properties

Product: \(\log_b(MN) = \log_b M + \log_b N\)  —  Expanded (sum) ↔ Condensed (product)

Quotient: \(\log_b\!\left(\dfrac{M}{N}\right) = \log_b M - \log_b N\)  —  Expanded (difference) ↔ Condensed (quotient)

Power: \(\log_b(M^n) = n\log_b M\)  —  Coefficient ↔ Exponent

Practice 10.8–10.10Log Property Practice

Expand each expression completely.

  1. \(\log_2(4x)\)
  2. \(\log(x^4 y)\)
  3. \(\ln\!\left(\dfrac{e^3}{x}\right)\)
  4. \(\log_5\!\left(\dfrac{x^2}{25}\right)\)
  5. \(\log\!\left(\dfrac{x^3 y^2}{z^4}\right)\)
  6. \(\ln\!\left(\sqrt{x}\right)\)
  7. \(\log_3\!\left(9x^5\right)\)
  8. \(\log_b\!\left(\dfrac{x^2 y^3}{w}\right)\)

Condense each expression into a single logarithm.

  1. \(\log(5) + \log(x)\)
  2. \(3\log_2(x) - \log_2(y)\)
  3. \(\ln(x) + \ln(4) + 2\ln(y)\)
  4. \(4\log(x) - 3\log(y) - \log(z)\)
  5. \(\tfrac{1}{2}\ln(x+1) + \ln(3)\)
  6. \(\log_5(x) - 2\log_5(3) + \log_5(y)\)
Sections 10.11 – 10.12
Solving Exponential and Logarithmic Equations
Exponential Equations · Logarithmic Equations · Extraneous Solutions
Section 10.11

Solving Exponential Equations

In Chapter 9, you solved some exponential equations by writing both sides as powers of the same base. For example, \(2^x = 32\) is solved by recognizing \(32 = 2^5\), so \(x = 5\). But this technique only works when both sides can be written as the same base. For equations like \(2^x = 50\), logarithms provide the solution.

Strategy for Solving Exponential Equations

1. Isolate the exponential expression on one side if necessary.

2. Take the logarithm of both sides (use \(\log\) or \(\ln\) — whichever matches the base or is more convenient).

3. Apply the Power Property to bring the exponent down.

4. Solve the resulting equation.

5. Check the solution in the original equation.

Example 10.11.1
Solving a basic exponential equation
  • ProblemSolve \(3^x = 50\). Round to three decimal places.
  • Think50 is not a simple power of 3. Take the common logarithm of both sides and use the Power Property.
  • Solve \[3^x = 50\] \[\log(3^x) = \log(50)\] \[x \cdot \log(3) = \log(50)\] \[x = \frac{\log(50)}{\log(3)} = \frac{1.69897}{0.47712} \approx 3.561\]
  • Check\(3^{3.561} \approx 50\) ✓
Answer: \(x \approx 3.561\)
Example 10.11.2
Solving with base \(e\)
  • ProblemSolve \(e^{2x} = 15\). Round to three decimal places.
  • ThinkThe base is \(e\), so take the natural logarithm of both sides. \(\ln(e^{2x}) = 2x\) because \(\ln\) and \(e^x\) are inverses.
  • Solve \[\ln(e^{2x}) = \ln(15) \;\Rightarrow\; 2x = \ln(15) \;\Rightarrow\; x = \frac{\ln(15)}{2} \approx \frac{2.708}{2} \approx 1.354\]
  • Check\(e^{2(1.354)} = e^{2.708} \approx 15\) ✓
Answer: \(x \approx 1.354\)
Example 10.11.3
Solving with isolation first
  • ProblemSolve \(5 \cdot 2^{x-1} = 80\).
  • ThinkFirst isolate the exponential part by dividing both sides by 5. Then recognize if the result is a perfect power.
  • Solve \[5 \cdot 2^{x-1} = 80 \;\Rightarrow\; 2^{x-1} = 16 = 2^4 \;\Rightarrow\; x - 1 = 4 \;\Rightarrow\; x = 5\]
  • Check\(5 \cdot 2^{5-1} = 5 \cdot 16 = 80\) ✓
Answer: \(x = 5\)
Example 10.11.4
Exponential decay application
  • ProblemA radioactive substance decays according to the model \(A(t) = 500 \cdot e^{-0.04t}\), where \(A\) is the amount in grams and \(t\) is time in years. How long does it take to decay to 100 grams?
  • ThinkSet \(A(t) = 100\) and solve for \(t\).
  • Solve \[500 \cdot e^{-0.04t} = 100 \;\Rightarrow\; e^{-0.04t} = 0.2\] \[\ln(e^{-0.04t}) = \ln(0.2) \;\Rightarrow\; -0.04t = \ln(0.2) \approx -1.6094\] \[t = \frac{-1.6094}{-0.04} \approx 40.24 \text{ years}\]
  • Check\(500 \cdot e^{-0.04(40.24)} \approx 500 \cdot 0.1998 \approx 100\) ✓
Answer: Approximately 40.2 years for the substance to decay to 100 grams.
Tip — Same-Base Shortcut

When both sides of an exponential equation can be written as the same base, try rewriting before using logarithms. This often gives an exact answer without a calculator. When no common base works, logarithms give a reliable decimal approximation.

Section 10.12

Solving Logarithmic Equations

Two Types of Logarithmic Equations

Type 1: A single logarithm on one side — convert to exponential form and solve.

Type 2: Logarithms on both sides — use properties to condense, then use the one-to-one property: if \(\log_b(M) = \log_b(N)\), then \(M = N\).

Critical: Always check solutions in the original equation. A solution that makes any logarithm argument equal to zero or negative is an extraneous solution and must be rejected.

Example 10.12.1
Solving by converting to exponential form
  • ProblemSolve \(\log_3(x + 2) = 4\).
  • ThinkSingle logarithm on the left. Convert to exponential form and solve for \(x\).
  • Solve \[3^4 = x + 2 \;\Rightarrow\; 81 = x + 2 \;\Rightarrow\; x = 79\]
  • Check\(\log_3(79 + 2) = \log_3(81) = 4\) ✓ (since \(3^4 = 81\))
Answer: \(x = 79\)
Example 10.12.2
Solving with properties first — extraneous solution
  • ProblemSolve \(\log(x) + \log(x - 3) = 1\).
  • ThinkUse the Product Property to combine the two logarithms into one. Then convert to exponential form (base 10) and solve.
  • Solve \[\log\!\big(x(x-3)\big) = 1 \;\Rightarrow\; 10^1 = x(x-3) \;\Rightarrow\; x^2 - 3x - 10 = 0\] \[(x-5)(x+2) = 0 \;\Rightarrow\; x = 5 \quad\text{or}\quad x = -2\]
  • Check \(x = 5\): \(\log(5) + \log(2) = \log(10) = 1\) ✓
    \(x = -2\): \(\log(-2)\) is undefined. ✗ — Extraneous. Reject.
Answer: \(x = 5\)
Example 10.12.3
Solving with logarithms on both sides
  • ProblemSolve \(\ln(x + 4) = \ln(2x - 1)\).
  • ThinkBoth sides are natural logarithms. Use the one-to-one property: set the arguments equal.
  • Solve \[x + 4 = 2x - 1 \;\Rightarrow\; 5 = x\]
  • Check\(\ln(5+4) = \ln(9)\) and \(\ln(2\cdot5-1) = \ln(9)\) ✓. Both arguments positive. ✓
Answer: \(x = 5\)
Example 10.12.4
Solving using the Quotient Property
  • ProblemSolve \(\log_2(x + 6) - \log_2(x) = 3\).
  • ThinkUse the Quotient Property to combine the left side into a single logarithm, then convert to exponential form.
  • Solve \[\log_2\!\left(\frac{x+6}{x}\right) = 3 \;\Rightarrow\; 2^3 = \frac{x+6}{x} \;\Rightarrow\; 8x = x + 6 \;\Rightarrow\; 7x = 6 \;\Rightarrow\; x = \frac{6}{7}\]
  • CheckBoth \(\frac{6}{7} > 0\) and \(\frac{6}{7} + 6 > 0\). ✓   \(\log_2\!\left(\frac{\frac{48}{7}}{\frac{6}{7}}\right) = \log_2(8) = 3\) ✓
Answer: \(x = \dfrac{6}{7}\)
Practice 10.11–10.12Solving Equations

Exponential equations — solve and show all steps. Round to three decimal places where needed.

  1. \(5^x = 625\)
  2. \(2^{3x} = 64\)
  3. \(3^x = 20\)
  4. \(e^x = 40\)
  5. \(4 \cdot e^{2x} = 100\)
  6. \(7^{x-2} = 50\)
  7. \(6 \cdot 2^x = 192\)
  8. \(e^{-0.05t} = 0.5\)

Logarithmic equations — solve and check for extraneous solutions.

  1. \(\log_2(x) = 5\)
  2. \(\log(x-3) = 2\)
  3. \(\ln(2x) = 6\)
  4. \(\log_4(x+1) = 3\)
  5. \(\log(x) + \log(x+9) = 1\)
  6. \(\ln(x+2) = \ln(3x-4)\)
  7. \(\log_3(x+6) - \log_3(x) = 2\)
  8. \(2\log_5(x) = \log_5(36)\)
Sections 10.13 – 10.14
Logarithmic Scales and Common Mistakes
Richter Scale · Decibels · pH · Seven Common Errors
Section 10.13

Logarithmic Scales

Some quantities vary over such enormous ranges that a standard linear scale becomes impractical. Earthquakes can release energies from trivially small tremors to catastrophic events — the difference is a factor of billions. Sound pressure varies from a barely audible whisper to a jet engine by a factor of trillions. In each case, mathematicians and scientists use logarithmic scales, which compress these vast ranges into manageable numbers by taking the logarithm of the quantity.

The Richter Scale

The Richter scale measures earthquake magnitude. The magnitude \(M\) is defined as:

\[M = \log\!\left(\frac{I}{I_0}\right)\]

where \(I\) is the intensity of the earthquake and \(I_0\) is a reference intensity (the smallest detectable earthquake). A magnitude 5 earthquake is not twice as intense as a magnitude 4 — it is ten times as intense, because the scale is logarithmic.

Example 10.13.1
Comparing earthquake magnitudes
  • ProblemAn earthquake measures 6.5 on the Richter scale. A second measures 4.5. How many times more intense is the first?
  • ThinkThe magnitudes differ by \(6.5 - 4.5 = 2\). On a base-10 logarithmic scale, a difference of 2 corresponds to a factor of \(10^2\) in intensity.
  • Solve\[\frac{I_1}{I_2} = 10^{6.5-4.5} = 10^2 = 100\]
Answer: The 6.5 earthquake is 100 times more intense than the 4.5 earthquake.

The Decibel Scale

The decibel (dB) measures sound intensity. The decibel level \(L\) is:

\[L = 10 \cdot \log\!\left(\frac{I}{I_0}\right)\]

where \(I\) is the sound intensity in watts per square meter and \(I_0 = 10^{-12}\) W/m² is the threshold of human hearing. A 10 dB increase corresponds to a factor of 10 in intensity; a 20 dB increase corresponds to a factor of 100.

Example 10.13.2
Computing decibel level
  • ProblemA sound has intensity \(I = 10^{-5}\) W/m². Find the decibel level.
  • Solve \[L = 10 \cdot \log\!\left(\frac{10^{-5}}{10^{-12}}\right) = 10 \cdot \log(10^7) = 10 \cdot 7 = 70 \text{ dB}\]
Answer: 70 dB — roughly equivalent to a vacuum cleaner or busy restaurant.

The pH Scale

In chemistry, the pH of a solution measures its acidity. It is defined as:

\[\text{pH} = -\log[\text{H}^+]\]

where \([\text{H}^+]\) is the concentration of hydrogen ions in moles per liter. A lower pH indicates a more acidic solution. A difference of 1 on the pH scale corresponds to a factor of 10 in hydrogen ion concentration.

Example 10.13.3
Computing pH
  • ProblemA solution has \([\text{H}^+] = 3.2 \times 10^{-4}\) mol/L. Find the pH.
  • Solve \[\text{pH} = -\log(3.2 \times 10^{-4}) = -[\log(3.2) + \log(10^{-4})] = -[0.505 + (-4)] = -(-3.495) \approx 3.5\]
Answer: pH ≈ 3.5. This is acidic (vinegar has a pH near 3).
MGU Library Connection
  • Logarithmic scales are discussed further in MGU Chemistry, Physics, and Environmental Science
  • The decibel scale is revisited in MGU Engineering courses on signal processing and acoustics
  • The pH scale appears in MGU Biology and Environmental Studies
  • Seismology and earthquake measurement are covered in MGU Earth Science
Practice 10.13Logarithmic Scale Applications
  1. An earthquake measures 7.8 on the Richter scale. A second earthquake measures 5.8. How many times more intense is the first?
  2. A sound has intensity \(10^{-9}\) W/m². Find the decibel level. (Use \(I_0 = 10^{-12}\) W/m².)
  3. A solution has hydrogen ion concentration \([\text{H}^+] = 5 \times 10^{-6}\) mol/L. Find the pH.
  4. A solution has pH 8.4. Find the hydrogen ion concentration.
  5. Two sounds measure 60 dB and 85 dB. How many times more intense is the louder sound?
  6. An earthquake measuring 6.0 occurs. What would be the magnitude of an earthquake ten times more intense?
Section 10.14

Common Logarithm Mistakes

Mistake 1 — Believing \(\log_b(M + N) = \log_b(M) + \log_b(N)\)
\(\log(3 + 7) = \log(3) + \log(7)\)

The Product Property applies to a product inside the logarithm, not a sum. \(\log(10) = 1\), but \(\log(3) + \log(7) = \log(21) \neq 1\). Correct form: \(\log(3 \cdot 7) = \log(3) + \log(7)\).

Mistake 2 — Believing \(\log_b(M - N) = \log_b(M) - \log_b(N)\)
\(\log(10 - 5) = \log(10) - \log(5)\)

The Quotient Property applies to a quotient inside the logarithm, not a difference of arguments. The correct form: \(\log\!\left(\frac{10}{5}\right) = \log(10) - \log(5) = \log(2)\). But \(\log(10 - 5) = \log(5) \neq \log(2)\).

Mistake 3 — Moving a coefficient as a multiplier, not as an exponent
\(3\log(x) = \log(3x)\)

The Power Property says \(n\log_b(M) = \log_b(M^n)\), not \(\log_b(nM)\). The coefficient moves to the exponent of the argument, not to a multiplier inside. Correct: \(3\log(x) = \log(x^3)\).

Mistake 4 — Forgetting to check for extraneous solutions
Reporting both solutions of a logarithmic equation without checking

When solving logarithmic equations, always substitute solutions back into the original equation and verify that all logarithm arguments are strictly positive. A solution that makes any argument zero or negative must be rejected.

Mistake 5 — Confusing \([\log(x)]^2\) and \(\log(x^2)\)
"Squaring the log is the same as squaring the argument"

\([\log(x)]^2\) means the logarithm itself is squared. \(\log(x^2) = 2\log(x)\) by the Power Property. These are different expressions with different values for most inputs.

Mistake 6 — Believing \(\log_b(x)\) is defined for \(x = 0\) or \(x < 0\)
\(\log_b(0)\) is defined

\(\log_b(0)\) is undefined — there is no exponent that raises a positive base to obtain 0. \(\log_b(\text{negative number})\) is also undefined in the real number system.

Mistake 7 — Applying properties when bases are different
\(\log_2(x) + \log_3(y) = \log_2(xy)\)

The Product, Quotient, and Power Properties only apply when all logarithms share the same base. Logarithms with different bases cannot be combined using these properties.

Chapter 10 — Back Matter

Chapter Summary

Key Terms Review

logarithm
The exponent to which the base must be raised to produce a given value. If \(b^y = x\), then \(\log_b(x) = y\).
base
The fixed positive number (not 1) that is raised to a power; the subscript in logarithmic notation.
argument
The input to a logarithmic function; the number inside the logarithm. Must be strictly positive.
logarithmic form
An equation written using logarithm notation: \(\log_b(x) = y\). The exponential form of the same equation is \(b^y = x\).
common logarithm
The logarithm base 10. Written \(\log(x)\) without a subscript. Used in scientific scales and order-of-magnitude reasoning.
natural logarithm
The logarithm base \(e \approx 2.71828\). Written \(\ln(x)\). Used in continuous growth, decay, and calculus.
logarithmic function
A function of the form \(f(x) = \log_b(x)\); the inverse of the exponential function \(g(x) = b^x\).
change of base
The formula \(\log_b(x) = \frac{\log(x)}{\log(b)}\), allowing any logarithm to be evaluated using base 10 or base \(e\).
Product Property
\(\log_b(MN) = \log_b M + \log_b N\). The logarithm of a product equals the sum of the logarithms.
Quotient Property
\(\log_b(M/N) = \log_b M - \log_b N\). The logarithm of a quotient equals the difference of the logarithms.
Power Property
\(\log_b(M^n) = n\log_b M\). An exponent inside the argument of a logarithm can be moved to the front as a coefficient.
one-to-one property
If \(\log_b(M) = \log_b(N)\), then \(M = N\). A logarithmic function assigns distinct outputs to distinct inputs.
extraneous solution
A value produced algebraically that does not satisfy the original equation, typically because it makes a logarithm argument non-positive.
logarithmic scale
A scale based on logarithms, used when quantities span many orders of magnitude. Each unit represents a multiplicative change.
decibel
The unit of sound intensity level. Defined as \(L = 10\log(I/I_0)\). A 10 dB increase corresponds to a tenfold increase in intensity.
pH
A measure of acidity defined as \(\text{pH} = -\log[\text{H}^+]\). Lower pH indicates a more acidic solution.
Richter scale
A logarithmic scale measuring earthquake magnitude. A difference of 1 in magnitude corresponds to a factor of 10 in intensity.

Concept Review Questions

Answer each question in one to three sentences without performing extensive computation.

  1. Explain in your own words what \(\log_b(x)\) asks.
  2. Why is the argument of a logarithm restricted to positive numbers?
  3. What is the connection between \(y = b^x\) and \(y = \log_b(x)\)? Describe their relationship geometrically.
  4. What is the domain and range of a logarithmic function? How do these compare to the exponential function?
  5. Explain the difference between \(\log(x)\) and \(\ln(x)\).
  6. Without computing, explain why \(\log_5(5) = 1\) and \(\log_5(1) = 0\).
  7. What is the vertical asymptote of the graph of \(y = \log_b(x)\), and why does it exist?
  8. In your own words, state the Product Property of Logarithms and explain why it works.
  9. Explain the difference between \(\log_b(M - N)\) and \(\log_b(M) - \log_b(N)\).
  10. What is an extraneous solution, and why do logarithmic equations sometimes produce them?
  11. Using the Richter scale as an example, explain what a difference of 3 in magnitude means in terms of earthquake intensity.
  12. How does taking the natural logarithm of both sides help solve an exponential equation involving base \(e\)?

Skill Practice

Part ALogarithm Evaluation

Evaluate without a calculator. Show the exponential form.

  1. \(\log_2(16)\)
  2. \(\log_3(27)\)
  3. \(\log_5(125)\)
  4. \(\log_4(64)\)
  5. \(\log_7(1)\)
  6. \(\log_6(6)\)
  7. \(\log(0.001)\)
  8. \(\log_2\!\left(\tfrac{1}{8}\right)\)
  9. \(\ln(e^4)\)
  10. \(\ln(1)\)
  11. \(\log_9(3)\)
  12. \(\log_4(8)\)
Part BExpanding and Condensing

Expand completely.

  1. \(\log_2(4x)\)
  2. \(\log(x^4 y)\)
  3. \(\ln\!\left(\dfrac{e^3}{x}\right)\)
  4. \(\log_5\!\left(\dfrac{x^2}{25}\right)\)
  5. \(\log\!\left(\dfrac{x^3 y^2}{z^4}\right)\)
  6. \(\ln\!\left(\sqrt{x}\right)\)

Condense into a single logarithm.

  1. \(\log(5) + \log(x)\)
  2. \(3\log_2(x) - \log_2(y)\)
  3. \(\ln(x) + \ln(4) + 2\ln(y)\)
  4. \(4\log(x) - 3\log(y) - \log(z)\)
  5. \(\tfrac{1}{2}\ln(x+1) + \ln(3)\)
  6. \(\log_5(x) - 2\log_5(3) + \log_5(y)\)
Part CEquation Solving
  1. \(5^x = 625\)
  2. \(3^x = 20\)
  3. \(e^x = 40\)
  4. \(4e^{2x} = 100\)
  5. \(\log_2(x) = 5\)
  6. \(\log(x-3) = 2\)
  7. \(\log(x)+\log(x+9)=1\)
  8. \(\log_3(x+6)-\log_3(x)=2\)

Word Problems

For each problem: set up the equation, solve, check, and answer in context.

  1. A population grows according to \(P(t) = 2000 \cdot e^{0.03t}\), where \(t\) is years. When will the population reach 8,000? Round to one decimal place.
  2. A bacteria colony doubles every 4 hours. Starting with 200 bacteria, how long until there are 25,600? Use logarithms to solve.
  3. A radioactive substance decays by the model \(A(t) = 800 \cdot e^{-0.025t}\). How long until only 100 grams remain?
  4. An investment of $5,000 earns continuously compounded interest at 4%. Use \(A = Pe^{rt}\) to find how long until the investment doubles.
  5. A sound registers 45 dB. A second sound is 1,000 times more intense. What is the decibel level of the second sound?
  6. A solution has \([\text{H}^+] = 2.5 \times 10^{-9}\) mol/L. Calculate the pH and classify as acidic, neutral, or basic.

Error Analysis

Identify the error and write the correct solution.

Problem 1
\(\log(x + y) = \log(x) + \log(y)\)

What property rule did this student confuse? What is the correct form using the Product Property?

Problem 2
Solving \(\log_3(x-4) + \log_3(x) = 2\) gives \(x = 9\) and \(x = -1\). Student reports both.

Is \(x = -1\) valid? Show the check and explain why or why not.

Problem 3
\(4\log_2(x) = \log_2(4x)\)

What property did the student confuse? Write the correct application of the Power Property.

Problem 4
A student solves \(2^x = 30\) and writes \(x = \log_2(30)\), then computes \(x \approx \frac{\log(2)}{\log(30)}\).

What is wrong with this Change of Base application? Write the correct calculation.

Problem 5
A student graphs \(y = \log(x + 5)\) with a vertical asymptote at \(x = 0\).

What mistake did they make? What is the correct vertical asymptote and domain?

Chapter 10 Checkpoint

Chapter 10 Checkpoint
Complete without assistance — assesses all key skills and concepts from Chapter 10
Part A — Evaluation and Notation
  1. Convert \(6^3 = 216\) to logarithmic form.
  2. Convert \(\log_4(64) = 3\) to exponential form.
  3. Evaluate \(\log_2(32)\) without a calculator.
  4. Evaluate \(\log_{1/3}(9)\).
  5. Evaluate \(\ln(e^{-4})\).
  6. Use the Change of Base Formula to evaluate \(\log_6(200)\) to three decimal places.
Part B — Properties
  1. Expand \(\log_3(27x^4)\) completely.
  2. Expand \(\ln\!\left(\dfrac{x^5}{e^2 y}\right)\) completely.
  3. Condense \(2\log(x) + \log(y) - 3\log(z)\) into a single logarithm.
  4. Condense \(\ln(4) + \frac{1}{2}\ln(x) - \ln(3)\) into a single logarithm.
Part C — Graphing
  1. Sketch the graph of \(y = \log_3(x-2)+1\). Label the vertical asymptote, x-intercept, and at least two additional points.
  2. State the domain, range, and vertical asymptote of \(y = \log(x+5)-3\).
Part D — Equations
  1. Solve \(4^x = 200\). Round to three decimal places.
  2. Solve \(3e^{0.5x} = 45\). Round to three decimal places.
  3. Solve \(\log_5(x+4) = 3\).
  4. Solve \(\ln(x) + \ln(x-2) = \ln(3)\). Check for extraneous solutions.
  5. Solve \(\log_2(x+8) - \log_2(x) = 4\).
Part E — Application
  1. A population grows according to \(P(t) = 2000 \cdot e^{0.03t}\), where \(t\) is years. When will the population reach 8,000? Round to one decimal place.
  2. A sound registers 45 dB. A second sound is 1,000 times more intense. What is the decibel level of the second sound?
  3. A solution has \([\text{H}^+] = 2.5 \times 10^{-9}\) mol/L. Calculate the pH and classify the solution as acidic, neutral, or basic.
Looking Ahead
Chapter 11: Sequences and Series

Chapter 10 developed logarithms as the inverse of exponential functions and showed how logarithms solve exponential equations. In Chapter 11, you will encounter another context where exponential thinking is essential.

Sequences and Series introduces the study of ordered patterns and accumulated sums. Geometric sequences — where each term is found by multiplying the previous term by a fixed ratio — follow exponential patterns. The sum of a geometric series connects to compound interest, which you studied in Chapter 9 using exponential models. As terms accumulate, questions arise about when a series converges, how a sum grows, and how sigma notation organizes repeated addition.

The connection between Chapter 10 and Chapter 11 is most visible in this type of problem: if you invest a fixed amount at compound interest each period for a number of years, the total grows according to a geometric series. Logarithms can then be used to find how long it takes for the accumulated total to exceed a target — combining sequences, exponential functions, and logarithms in a single model.

MGU Library Connection
  • Chapter 9: Exponential Functions — the source chapter; logarithms are its inverse
  • Chapter 11: Sequences and Series — geometric sequences connect to exponential models and logarithms
  • Algebra II Appendix I: Exponential and Logarithmic Rules — reference sheet for this chapter
  • Precalculus / Trigonometry: logarithms and exponentials reappear in inverse function analysis and advanced modeling
  • Calculus I: the derivative and integral of \(\ln(x)\) and \(e^x\) are foundational results; the natural logarithm has a special geometric definition as an area function
  • MGU Chemistry and Environmental Science: pH, buffer systems, and reaction rate laws use logarithms throughout
  • MGU Engineering: signal processing, decibel calculations, and Bode plots rely on logarithmic scaling
  • MGU Dictionary entries: logarithm, natural logarithm, common logarithm, Product Property, Quotient Property, Power Property, extraneous solution