The Central Idea
Suppose
\[ b^x=y, \]where
\[ b>0,\qquad b\ne1,\qquad y>0. \]The same relationship can be written logarithmically as
\[ \boxed{\log_b(y)=x}. \]These equations say exactly the same thing.
A Logarithm Answers an Exponent Question
Consider
\[ 2^3=8. \]The logarithmic version is
\[ \boxed{\log_2(8)=3}. \]The logarithm asks:
To what exponent must \(2\) be raised to produce \(8\)?
The answer is \(3\).
Exponential Form and Logarithmic Form
The conversion pattern is
\[ \boxed{b^x=y\quad\Longleftrightarrow\quad\log_b(y)=x}. \]The base stays the same.
The exponent becomes the logarithm's output.
The exponential result becomes the logarithm's input.
Reading Logarithmic Notation
The expression
\[ \log_b(y) \]is read
“log base \(b\) of \(y\)”.
It asks for the exponent \(x\) satisfying
\[ b^x=y. \]The Three Roles in a Logarithm
In
\[ \log_b(y)=x, \]- \(b\) is the base;
- \(y\) is the argument;
- \(x\) is the logarithmic value, which is the exponent.
The logarithm returns an exponent.
Example: Convert to Logarithmic Form
Rewrite
\[ 5^3=125 \]in logarithmic form.
The base is \(5\), the result is \(125\), and the exponent is \(3\).
Therefore,
\[ \boxed{\log_5(125)=3}. \]Example: Convert to Exponential Form
Rewrite
\[ \log_4(64)=3 \]in exponential form.
The base is \(4\), the logarithmic value \(3\) becomes the exponent, and the argument \(64\) becomes the result:
\[ \boxed{4^3=64}. \]Why Logarithms Are Inverses
An exponential function sends an exponent to a value:
\[ x\longmapsto b^x. \]The logarithm reverses that mapping:
\[ b^x\longmapsto x. \]Therefore,
\[ \boxed{\log_b(b^x)=x}. \]The Other Inverse Identity
If logarithms undo exponentiation, exponentiation must also undo logarithms.
Therefore,
\[ \boxed{b^{\log_b(x)}=x}, \]for \(x>0\).
The Two Fundamental Inverse Identities
\[ \boxed{\log_b(b^x)=x} \]and
\[ \boxed{b^{\log_b(x)}=x}, \]where
\[ b>0,\qquad b\ne1,\qquad x>0 \]where required.
Example: Using an Inverse Identity
Simplify
\[ \log_3(3^7). \]The logarithm base \(3\) undoes exponentiation by base \(3\):
\[ \boxed{\log_3(3^7)=7}. \]Example: Exponentiating a Logarithm
Simplify
\[ 5^{\log_5(12)}. \]Exponentiation by base \(5\) reverses the base-\(5\) logarithm:
\[ \boxed{5^{\log_5(12)}=12}. \]Evaluating Logarithms by Recognition
To evaluate
\[ \log_2(32), \]ask:
What exponent on \(2\) produces \(32\)?
Since
\[ 2^5=32, \]we have
\[ \boxed{\log_2(32)=5}. \]Logarithms Can Be Zero
Consider
\[ \log_b(1). \]We ask:
What exponent on \(b\) gives \(1\)?
Since
\[ b^0=1, \]we obtain
\[ \boxed{\log_b(1)=0}. \]The Logarithm of the Base
Consider
\[ \log_b(b). \]Since
\[ b^1=b, \]we obtain
\[ \boxed{\log_b(b)=1}. \]Logarithms Can Be Negative
Consider
\[ \log_2\left(\frac18\right). \]Since
\[ 2^{-3}=\frac18, \]we have
\[ \boxed{\log_2\left(\frac18\right)=-3}. \]A negative logarithmic value means the required exponent is negative.
Logarithms Can Be Fractional
Consider
\[ \log_9(3). \]Since
\[ 9^{1/2}=3, \]we obtain
\[ \boxed{\log_9(3)=\frac12}. \]Example: A Rational Logarithmic Value
Evaluate
\[ \log_8(4). \]We need \(x\) such that
\[ 8^x=4. \]Rewrite both quantities using base \(2\):
\[ (2^3)^x=2^2. \]Thus
\[ 3x=2. \]Therefore,
\[ \boxed{\log_8(4)=\frac23}. \]Why the Logarithmic Base Must Be Positive
A standard real logarithm is defined as the inverse of
\[ b^x. \]To obtain a continuous real-valued exponential function defined for every real \(x\), we require
\[ \boxed{b>0}. \]Why the Base Cannot Be \(1\)
If
\[ b=1, \]then
\[ 1^x=1 \]for every \(x\).
The function would not be one-to-one and therefore would not have an inverse function.
Thus a logarithmic base must satisfy
\[ \boxed{b>0,\qquad b\ne1}. \]Why the Logarithm's Argument Must Be Positive
For a valid logarithmic base \(b\),
\[ b^x>0 \]for every real \(x\).
Therefore, the exponential function never produces zero or a negative number.
Since a logarithm reverses an exponential function, its real-valued input must satisfy
\[ \boxed{x>0}. \]Why \(\log_b(0)\) Is Undefined
Suppose
\[ \log_b(0)=x. \]This would mean
\[ b^x=0. \]But for \(b>0\),
\[ b^x>0 \]for every real \(x\).
Therefore,
\[ \boxed{\log_b(0)\text{ is undefined in the real numbers}}. \]Why Logarithms of Negative Numbers Are Not Real
Suppose
\[ \log_b(-5)=x. \]Then we would need
\[ b^x=-5. \]But a positive base raised to a real exponent remains positive.
Therefore,
\[ \boxed{\log_b(-5)\text{ is not a real number}}. \]Domain and Range of an Exponential Function
For
\[ f(x)=b^x, \]the domain is
\[ \boxed{(-\infty,\infty)} \]and the range is
\[ \boxed{(0,\infty)}. \]Domain and Range of a Logarithmic Function
For
\[ g(x)=\log_b(x), \]the domain is
\[ \boxed{(0,\infty)} \]and the range is
\[ \boxed{(-\infty,\infty)}. \]The domain and range swap because the two functions are inverses.
Inverse Functions Swap Inputs and Outputs
If
\[ f(3)=8, \]then the inverse satisfies
\[ f^{-1}(8)=3. \]For
\[ f(x)=2^x, \]the inverse is
\[ f^{-1}(x)=\log_2(x). \]Thus
\[ 2^3=8 \]corresponds to
\[ \log_2(8)=3. \]Points on Inverse Graphs Swap Coordinates
If
\[ (a,b) \]lies on a function, then
\[ (b,a) \]lies on its inverse.
Since
\[ (3,8) \]lies on
\[ y=2^x, \]the point
\[ (8,3) \]lies on
\[ y=\log_2(x). \]Graphs of Inverse Functions
A function and its inverse are reflections across the line
\[ \boxed{y=x}. \]Therefore, the graphs of
\[ y=b^x \]and
\[ y=\log_b(x) \]are mirror images across \(y=x\).
The Exponential Horizontal Asymptote Becomes a Logarithmic Vertical Asymptote
The basic exponential graph
\[ y=b^x \]has horizontal asymptote
\[ y=0. \]Reflecting across \(y=x\) turns this into the logarithmic vertical asymptote
\[ \boxed{x=0}. \]The Point \((0,1)\) Becomes \((1,0)\)
Every basic exponential function satisfies
\[ b^0=1. \]Therefore, every graph of
\[ y=b^x \]passes through
\[ (0,1). \]Its inverse logarithmic graph passes through the reflected point
\[ \boxed{(1,0)}. \]Increasing Logarithmic Functions
If
\[ b>1, \]then the exponential function
\[ y=b^x \]is increasing.
Its inverse
\[ y=\log_b(x) \]is also increasing.
Decreasing Logarithmic Functions
If
\[ 0is decreasing.
Therefore,
\[ y=\log_b(x) \]is also decreasing.
Example: Evaluate by Converting Forms
Evaluate
\[ \log_7(49). \]Write the equivalent exponential equation:
\[ 7^x=49. \]Since
\[ 7^2=49, \]we obtain
\[ \boxed{\log_7(49)=2}. \]Example: Evaluate a Negative Logarithmic Value
Evaluate
\[ \log_5\left(\frac1{125}\right). \]Since
\[ \frac1{125}=5^{-3}, \]we have
\[ \boxed{\log_5\left(\frac1{125}\right)=-3}. \]Example: Evaluate a Fractional Logarithmic Value
Evaluate
\[ \log_{16}(2). \]We seek \(x\) such that
\[ 16^x=2. \]Since
\[ 16=2^4, \]we have
\[ 2^{4x}=2^1. \]Therefore,
\[ 4x=1 \]and
\[ \boxed{\log_{16}(2)=\frac14}. \]Solving a Simple Exponential Equation by Recognition
Solve
\[ 3^x=81. \]Since
\[ 81=3^4, \]the equal bases imply
\[ \boxed{x=4}. \]Solving the Same Equation Logarithmically
Starting with
\[ 3^x=81, \]rewrite in logarithmic form:
\[ \boxed{x=\log_3(81)}. \]Since \(\log_3(81)=4\),
\[ \boxed{x=4}. \]Solving a Simple Logarithmic Equation
Solve
\[ \log_2(x)=5. \]Rewrite in exponential form:
\[ 2^5=x. \]Therefore,
\[ \boxed{x=32}. \]Example: Solve with a Negative Logarithmic Output
Solve
\[ \log_4(x)=-2. \]Rewrite:
\[ 4^{-2}=x. \]Since
\[ 4^{-2}=\frac1{16}, \]we obtain
\[ \boxed{x=\frac1{16}}. \]Logarithms Let Us Name Unknown Exponents
Consider
\[ 2^x=10. \]The value \(10\) is not an obvious integer power of \(2\).
Logarithmic notation gives an exact name to the unknown exponent:
\[ \boxed{x=\log_2(10)}. \]A calculator can later approximate this value.
Exact Values and Approximate Values
The expression
\[ \log_2(10) \]is an exact value.
A decimal such as
\[ 3.3219\ldots \]is an approximation.
Exact logarithmic notation is useful even when no simple rational exponent exists.
Logarithms and Doubling Time
Suppose a growth model has factor \(b>1\).
Doubling time \(D\) satisfies
\[ b^D=2. \]Therefore,
\[ \boxed{D=\log_b(2)}. \]This is the direct logarithmic meaning of doubling time.
Logarithms and Half-Life
Suppose a decay model has factor
\[ 0Therefore,
\[ \boxed{H=\log_b\left(\frac12\right)}. \]Why Logarithms Matter in Exponential Models
Exponential models naturally place time or another unknown in the exponent.
For example,
\[ 500(1.08)^t=1000 \]cannot be solved by ordinary subtraction, multiplication, division, or radicals alone.
A logarithm is the inverse operation that allows the exponent to be recovered.
Inverse Functions and Composition
If two functions are inverses, composing them returns the original input.
Let
\[ f(x)=b^x \]and
\[ g(x)=\log_b(x). \]Then
\[ g(f(x)) = \log_b(b^x) = x. \]Thus,
\[ \boxed{g\circ f=\operatorname{id}}. \]The Reverse Composition
Likewise,
\[ f(g(x)) = b^{\log_b(x)} = x, \]for \(x>0\).
Therefore,
\[ \boxed{f\circ g=\operatorname{id}} \]on the logarithm's domain.
Logarithmic Functions Are One-to-One
Because a logarithmic function is an inverse of a one-to-one exponential function, it is also one-to-one.
Therefore, if
\[ \log_b(x)=\log_b(y), \]then
\[ \boxed{x=y}, \]provided \(x>0\) and \(y>0\).
Exponential Functions Are One-to-One
Similarly, for a valid exponential base,
\[ b^x=b^y \]implies
\[ \boxed{x=y}. \]This property allows equations with matching exponential bases to be solved by equating exponents.
Common Error: Reversing the Wrong Parts
From
\[ 2^5=32, \]the correct logarithmic form is
\[ \boxed{\log_2(32)=5}. \]The base remains \(2\).
The exponent \(5\) becomes the logarithmic output, and \(32\) becomes the argument.
Common Error: Thinking \(\log_b(x)\) Means \(b/x\)
Logarithmic notation does not represent division.
The expression
\[ \log_b(x) \]asks for the exponent that makes
\[ b^{(\text{that exponent})}=x. \]It is an inverse exponent operation, not a fraction.
Common Error: Treating the Base as the Output
In
\[ \log_3(81)=4, \]the logarithm's output is \(4\), not \(3\).
The number \(3\) is the base.
The result \(4\) is the exponent required to produce \(81\).
Common Error: Taking the Logarithm of Zero
The expression
\[ \log_b(0) \]is undefined in the real numbers.
No real exponent on a positive base produces zero.
Common Error: Taking a Real Logarithm of a Negative Number
The expression
\[ \log_b(-7) \]has no real value for a standard real logarithmic base.
The argument of a real logarithm must be positive.
Common Error: Confusing \(\log_b(x)\) with \(b^x\)
These are inverse operations.
For example,
\[ 2^3=8 \]while
\[ \log_2(8)=3. \]One sends exponent \(3\) to value \(8\). The other sends value \(8\) back to exponent \(3\).
Common Error: Confusing an Inverse Function with a Reciprocal
If
\[ f(x)=2^x, \]then its inverse function is
\[ f^{-1}(x)=\log_2(x). \]This is not the same as the reciprocal
\[ \frac1{f(x)}=2^{-x}. \]Inverse functions undo each other's input-output mapping. Reciprocals undo multiplication.
A Reliable Procedure for Converting Exponential to Logarithmic Form
- Identify the exponential base.
- Keep that same number as the logarithmic base.
- Move the exponential result into the logarithm's argument.
- Move the exponent to the other side as the logarithmic value.
- Check by converting back to exponential form.
The pattern is
\[ \boxed{b^x=y\Longleftrightarrow\log_b(y)=x}. \]A Reliable Procedure for Evaluating a Logarithm Exactly
- Write \(\log_b(y)=x\).
- Convert to exponential form \(b^x=y\).
- Express \(y\) as a power of \(b\) when possible.
- Identify the exponent.
- Check the domain condition \(y>0\).
A Reliable Procedure for Solving a Basic Logarithmic Equation
- Identify the logarithmic base.
- Convert the equation to exponential form.
- Evaluate the resulting power.
- Solve for the unknown.
- Check that any logarithmic argument in the original equation is positive.
Example: Full Conversion Workflow
Rewrite
\[ 7^{-2}=\frac1{49} \]in logarithmic form.
Base:
\[ 7. \]Exponential result:
\[ \frac1{49}. \]Exponent:
\[ -2. \]Therefore,
\[ \boxed{\log_7\left(\frac1{49}\right)=-2}. \]Example: Full Evaluation Workflow
Evaluate
\[ \log_{25}(5). \]Let
\[ \log_{25}(5)=x. \]Convert to exponential form:
\[ 25^x=5. \]Rewrite \(25\) as \(5^2\):
\[ (5^2)^x=5^1. \]Therefore,
\[ 2x=1 \]and
\[ \boxed{x=\frac12}. \]Practice 1: Convert to Logarithmic Form
Rewrite
\[ 3^4=81 \]in logarithmic form.
Practice 2: Convert to Exponential Form
Rewrite
\[ \log_5(625)=4 \]in exponential form.
Practice 3: Evaluate a Logarithm
Evaluate
\[ \log_2(64). \]Practice 4: Evaluate a Negative Logarithmic Value
Evaluate
\[ \log_3\left(\frac1{27}\right). \]Practice 5: Evaluate a Fractional Logarithmic Value
Evaluate
\[ \log_{16}(4). \]Practice 6: Use an Inverse Identity
Simplify
\[ \log_7(7^{12}). \]Practice 7: Use the Reverse Inverse Identity
Simplify
\[ 4^{\log_4(19)}. \]Practice 8: Solve a Logarithmic Equation
Solve
\[ \log_3(x)=5. \]Practice 9: Solve an Exponential Equation
Solve
\[ 5^x=125. \]Practice 10: Identify the Domain Error
A student claims
\[ \log_2(-8)=3. \]Explain why the statement is invalid in the real numbers.
Practice Answers
Practice 1
\[ 3^4=81 \]becomes
\[ \boxed{\log_3(81)=4}. \]Practice 2
\[ \log_5(625)=4 \]becomes
\[ \boxed{5^4=625}. \]Practice 3
Since
\[ 2^6=64, \]we have
\[ \boxed{\log_2(64)=6}. \]Practice 4
Since
\[ 3^{-3}=\frac1{27}, \]we have
\[ \boxed{\log_3\left(\frac1{27}\right)=-3}. \]Practice 5
Since
\[ 16^{1/2}=4, \]we have
\[ \boxed{\log_{16}(4)=\frac12}. \]Practice 6
\[ \boxed{\log_7(7^{12})=12}. \]Practice 7
\[ \boxed{4^{\log_4(19)}=19}. \]Practice 8
\[ \log_3(x)=5 \]becomes
\[ 3^5=x. \]Therefore,
\[ \boxed{x=243}. \]Practice 9
Since
\[ 125=5^3, \]we have
\[ 5^x=5^3. \]Therefore,
\[ \boxed{x=3}. \]Practice 10
A real logarithm requires a positive argument.
The statement
\[ \log_2(-8)=3 \]would imply
\[ 2^3=-8, \]but
\[ 2^3=8. \]Therefore,
\[ \boxed{\log_2(-8)\text{ is not defined in the real numbers}}. \]What to Remember
Exponential and logarithmic forms are equivalent:
\[ \boxed{b^x=y\Longleftrightarrow\log_b(y)=x}. \]- A logarithm returns an exponent.
- The logarithmic base is the same as the exponential base.
- A valid real logarithmic base satisfies \(b>0\) and \(b\ne1\).
- The argument of a real logarithm must be positive.
- \(\log_b(1)=0\).
- \(\log_b(b)=1\).
- \(\log_b(b^x)=x\).
- \(b^{\log_b(x)}=x\) for \(x>0\).
- Exponential and logarithmic functions are inverses and reflect across \(y=x\).
- The exponential range becomes the logarithmic domain.
The most important question to ask when reading a logarithm is: what exponent on this base produces the argument?
Where We Go Next
Logarithms can use many different bases, but two bases occur so frequently that they receive special notation and special names.
The next lesson, Common and Natural Logarithms, develops base-\(10\) logarithms, base-\(e\) logarithms, calculator notation, and the central role of the natural logarithm in continuous exponential models.