The Central Idea
The common logarithm is logarithm base \(10\):
\[ \boxed{\log(x)=\log_{10}(x)}. \]The natural logarithm is logarithm base \(e\):
\[ \boxed{\ln(x)=\log_e(x)}. \]Both answer the same kind of question:
What exponent on the base produces the argument?
The Common Logarithm
The notation
\[ \log(x) \]conventionally means
\[ \log_{10}(x). \]Therefore,
\[ \log(x)=y \]means
\[ \boxed{10^y=x}. \]Example: Common Logarithm of \(1000\)
Evaluate
\[ \log(1000). \]Since
\[ 10^3=1000, \]we have
\[ \boxed{\log(1000)=3}. \]Common Logarithms of Powers of Ten
\[ \begin{array}{c|c} x & \log(x)\\ \hline 0.001 & -3\\ 0.01 & -2\\ 0.1 & -1\\ 1 & 0\\ 10 & 1\\ 100 & 2\\ 1000 & 3 \end{array} \]Each factor of \(10\) in the input changes the logarithm by \(1\).
Negative Common Logarithms
Consider
\[ \log(0.01). \]Since
\[ 0.01=10^{-2}, \]we obtain
\[ \boxed{\log(0.01)=-2}. \]The Natural Logarithm
The notation
\[ \ln(x) \]means
\[ \log_e(x). \]Therefore,
\[ \ln(x)=y \]means
\[ \boxed{e^y=x}. \]The Number \(e\)
The natural exponential base is
\[ \boxed{e\approx2.718281828\ldots}. \]It arises naturally in continuous growth, continuous compounding, differential equations, probability, and calculus.
The natural logarithm is the inverse of exponentiation with this base.
Example: \(\ln(e^4)\)
Evaluate
\[ \ln(e^4). \]Because \(\ln\) undoes exponentiation by \(e\),
\[ \boxed{\ln(e^4)=4}. \]Example: \(e^{\ln 7}\)
Evaluate
\[ e^{\ln7}. \]Exponentiation by \(e\) undoes the natural logarithm:
\[ \boxed{e^{\ln7}=7}. \]The Two Natural-Log Inverse Identities
\[ \boxed{\ln(e^x)=x} \]and
\[ \boxed{e^{\ln x}=x}, \qquad x>0. \]These are the base-\(e\) versions of the general logarithmic inverse laws.
The Two Common-Log Inverse Identities
\[ \boxed{\log(10^x)=x} \]and
\[ \boxed{10^{\log x}=x}, \qquad x>0. \]These are the corresponding base-\(10\) inverse relationships.
Why Base \(10\) Is Called Common
Base \(10\) fits naturally with the decimal number system.
Powers of \(10\) organize place value, scientific notation, and orders of magnitude:
\[ \ldots,10^{-2},10^{-1},10^0,10^1,10^2,\ldots \]Common logarithms convert these multiplicative powers of \(10\) into their exponents.
Why Base \(e\) Is Called Natural
The base \(e\) is mathematically natural because exponential change using \(e\) interacts especially cleanly with continuous rates.
For example, the derivative of
\[ e^x \]is again
\[ e^x. \]This makes \(e\) and \(\ln\) central in calculus and continuous models.
Common and Natural Logarithms Are Both Ordinary Logarithms
The symbols
\[ \log \]and
\[ \ln \]do not define different kinds of operations.
They are logarithms with different bases:
\[ \boxed{\log=\log_{10}} \]and
\[ \boxed{\ln=\log_e}. \]Exact Values from Recognizable Powers
Evaluate
\[ \log(10^6). \]By inverse structure,
\[ \boxed{\log(10^6)=6}. \]Likewise,
\[ \boxed{\ln(e^{-3})=-3}. \]Values That Are Not Simple Powers
Many logarithms do not simplify to integers or simple rational numbers.
For example,
\[ \ln2 \]is the exact exponent that makes
\[ e^{\ln2}=2. \]Numerically,
\[ \boxed{\ln2\approx0.6931}. \]Another Irrational Logarithmic Value
Consider
\[ \log2. \]This is the exponent \(x\) satisfying
\[ 10^x=2. \]Numerically,
\[ \boxed{\log2\approx0.3010}. \]Exact Form versus Decimal Approximation
The expression
\[ \ln2 \]is exact.
The expression
\[ 0.6931 \]is an approximation.
Therefore, write
\[ \boxed{\ln2\approx0.6931} \]rather than using an equals sign for the rounded decimal.
Calculator Keys
Scientific calculators typically provide keys labeled
\[ \boxed{\log} \]and
\[ \boxed{\ln}. \]The \(\log\) key evaluates base-\(10\) logarithms, while the \(\ln\) key evaluates base-\(e\) logarithms.
Example: Calculator Evaluation of \(\log750\)
Since \(750\) is not an exact integer power of \(10\), use a calculator:
\[ \boxed{\log750\approx2.8751}. \]Example: Calculator Evaluation of \(\ln25\)
Since \(25\) is not an obvious power of \(e\),
\[ \boxed{\ln25\approx3.2189}. \]Inverse Calculator Keys
Calculators often pair
\[ \log \]with
\[ 10^x, \]and pair
\[ \ln \]with
\[ e^x. \]This pairing reflects the inverse-function relationship.
Example: Undo a Natural Logarithm
If
\[ \ln x=3, \]then exponentiate with base \(e\):
\[ x=e^3. \]Numerically,
\[ \boxed{x\approx20.0855}. \]Example: Undo a Common Logarithm
If
\[ \log x=2.5, \]then
\[ x=10^{2.5}. \]The common logarithm and \(10^x\) undo each other.
Domain of Common and Natural Logarithms
Both
\[ \log x \]and
\[ \ln x \]require
\[ \boxed{x>0}. \]Neither has a real value at \(x=0\) or for negative \(x\).
Range of Common and Natural Logarithms
Both logarithmic functions can return any real number.
Therefore, the range of
\[ y=\log x \]and
\[ y=\ln x \]is
\[ \boxed{(-\infty,\infty)}. \]The Point \((1,0)\)
Since
\[ 10^0=1 \]and
\[ e^0=1, \]we have
\[ \log1=0 \]and
\[ \ln1=0. \]Therefore, both graphs pass through
\[ \boxed{(1,0)}. \]Vertical Asymptote
Both
\[ y=\log x \]and
\[ y=\ln x \]have vertical asymptote
\[ \boxed{x=0}. \]Their graphs approach the \(y\)-axis from the right but never cross it.
Both Functions Are Increasing
Since
\[ 10>1 \]and
\[ e>1, \]both
\[ y=\log x \]and
\[ y=\ln x \]are increasing functions.
Common and Natural Logs Have Different Numerical Values
Even though both functions are logarithms, they use different bases.
For example,
\[ \log10=1 \]but
\[ \ln10\approx2.3026. \]The input is the same, but the exponent is measured relative to a different base.
Why Different Bases Give Different Answers
The expression
\[ \log10=1 \]because
\[ 10^1=10. \]The expression
\[ \ln10 \]asks for the exponent \(x\) such that
\[ e^x=10. \]Different bases require different exponents to reach the same value.
Common Logarithms and Orders of Magnitude
If
\[ 10^n\le x<10^{n+1}, \]then
\[ n\le\log xExample: Estimate before Calculating
Since
\[ 100<750<1000, \]we know
\[ 2<\log750<3. \]The calculator value
\[ \log750\approx2.8751 \]fits that estimate.
Natural Logarithms and Continuous Growth
Suppose
\[ A(t)=A_0e^{kt}. \]Divide by \(A_0\):
\[ \frac{A(t)}{A_0}=e^{kt}. \]Taking natural logarithms gives
\[ \boxed{ \ln\left(\frac{A(t)}{A_0}\right)=kt }. \]Solving Continuous Growth for Time
Continue from
\[ \ln\left(\frac{A}{A_0}\right)=kt. \]Divide by \(k\):
\[ \boxed{ t= \frac1k \ln\left(\frac{A}{A_0}\right) }. \]This is one reason \(\ln\) appears constantly in continuous-growth and continuous-decay formulas.
Natural Logarithms and Continuous Compounding
Continuous compounding uses
\[ A=Pe^{rt}. \]If time is unknown,
\[ \frac AP=e^{rt}. \]Taking natural logarithms gives
\[ \ln\left(\frac AP\right)=rt. \]Therefore,
\[ \boxed{ t= \frac1r\ln\left(\frac AP\right) }. \]Natural Logarithms and Doubling Time
For continuous growth,
\[ A=A_0e^{kt}. \]At doubling time \(D\),
\[ 2=e^{kD}. \]Taking \(\ln\),
\[ \ln2=kD. \]Therefore,
\[ \boxed{ D=\frac{\ln2}{k} }. \]Natural Logarithms and Half-Life
For continuous decay,
\[ A=A_0e^{-kt}. \]At half-life \(H\),
\[ \frac12=e^{-kH}. \]Therefore,
\[ \boxed{ H=\frac{\ln2}{k} }. \]Common Logarithms and Scientific Notation
Scientific notation writes a positive number as
\[ x=a\times10^n, \]where
\[ 1\le a<10. \]Taking the common logarithm gives
\[ \log x = \log a+n. \]This connects logarithms directly to powers of ten.
Historical Use of Common Logarithms
Before electronic calculators, tables of common logarithms were important computational tools.
Logarithm laws transform multiplication into addition and division into subtraction, making difficult numerical calculations easier to carry out by hand.
Modern calculators have changed the practical method, but the algebraic structure remains important.
Common Error: Assuming \(\log\) Has No Base
In standard algebra notation,
\[ \log x \]abbreviates
\[ \boxed{\log_{10}x}. \]The base is omitted because base \(10\) is understood by convention.
Common Error: Thinking \(\ln\) Means “Log with No Base”
The natural logarithm has a specific base:
\[ \boxed{\ln x=\log_e x}. \]It is not a logarithm with an unspecified base.
Common Error: Treating \(\log x\) and \(\ln x\) as Equal
In general,
\[ \boxed{\log x\ne\ln x}. \]They use different bases and therefore return different exponents.
Common Error: Using an Invalid Logarithmic Argument
Neither
\[ \log0 \]nor
\[ \ln0 \]is defined in the real numbers.
Likewise, real common and natural logarithms do not accept negative arguments.
Common Error: Confusing \(\ln x\) with \(1/x\)
The natural logarithm
\[ \ln x \]is not the reciprocal of \(x\).
It is the exponent satisfying
\[ e^{\ln x}=x. \]Common Error: Confusing \(e^x\) with \(ex\)
The expression
\[ e^x \]is exponential.
The expression
\[ ex \]is ordinary multiplication.
These are completely different operations.
Common Error: Rounding Too Early
Values such as
\[ \ln2 \]are irrational.
When they appear in larger calculations, retain calculator precision until the final result rather than repeatedly rounding intermediate values.
A Reliable Procedure for Evaluating \(\log x\)
- Check that \(x>0\).
- Ask whether \(x\) is a recognizable power of \(10\).
- If so, identify the exponent exactly.
- If not, use the calculator's \(\log\) function.
- Estimate the order of magnitude when useful as a reasonableness check.
- Round only as requested.
A Reliable Procedure for Evaluating \(\ln x\)
- Check that \(x>0\).
- Ask whether \(x\) is a recognizable power of \(e\).
- If so, identify the exponent exactly.
- If not, use the calculator's \(\ln\) function.
- Keep the logarithmic expression exact when an exact form is preferred.
- Round only at the final stage.
A Reliable Procedure for Undoing a Logarithm
If
\[ \log x=a, \]write
\[ \boxed{x=10^a}. \]If
\[ \ln x=a, \]write
\[ \boxed{x=e^a}. \]Always preserve the original logarithmic domain.
Example: Full Common-Log Workflow
Evaluate
\[ \log(0.0001). \]Rewrite the input as a power of \(10\):
\[ 0.0001=10^{-4}. \]Therefore,
\[ \boxed{\log(0.0001)=-4}. \]Example: Full Natural-Log Workflow
Evaluate
\[ \ln(e^{5/2}). \]The natural logarithm undoes exponentiation by \(e\):
\[ \boxed{ \ln(e^{5/2})=\frac52 }. \]Practice 1: Common Logarithm
Evaluate
\[ \log(10000). \]Practice 2: Negative Common Logarithm
Evaluate
\[ \log(0.001). \]Practice 3: Natural Logarithm
Evaluate
\[ \ln(e^6). \]Practice 4: Natural-Log Inverse
Simplify
\[ e^{\ln12}. \]Practice 5: Common-Log Inverse
Simplify
\[ 10^{\log35}. \]Practice 6: Approximate a Common Logarithm
Approximate
\[ \log40. \]Practice 7: Approximate a Natural Logarithm
Approximate
\[ \ln7. \]Practice 8: Solve with a Common Logarithm
Solve
\[ \log x=3. \]Practice 9: Solve with a Natural Logarithm
Solve
\[ \ln x=4. \]Practice 10: Domain Check
State the real domain of
\[ f(x)=\ln(x-5). \]Practice Answers
Practice 1
Since
\[ 10000=10^4, \]we have
\[ \boxed{\log(10000)=4}. \]Practice 2
Since
\[ 0.001=10^{-3}, \]we have
\[ \boxed{\log(0.001)=-3}. \]Practice 3
\[ \boxed{\ln(e^6)=6}. \]Practice 4
\[ \boxed{e^{\ln12}=12}. \]Practice 5
\[ \boxed{10^{\log35}=35}. \]Practice 6
\[ \boxed{\log40\approx1.6021}. \]Practice 7
\[ \boxed{\ln7\approx1.9459}. \]Practice 8
\[ \log x=3 \]means
\[ x=10^3. \]Therefore,
\[ \boxed{x=1000}. \]Practice 9
\[ \ln x=4 \]means
\[ \boxed{x=e^4}. \]Practice 10
The logarithmic argument must be positive:
\[ x-5>0. \]Therefore,
\[ \boxed{x>5}. \]The domain is
\[ \boxed{(5,\infty)}. \]What to Remember
The two special logarithms are
\[ \boxed{\log x=\log_{10}x} \]and
\[ \boxed{\ln x=\log_e x}. \]- Common logarithms use base \(10\).
- Natural logarithms use base \(e\).
- \(\log(10^x)=x\).
- \(10^{\log x}=x\) for \(x>0\).
- \(\ln(e^x)=x\).
- \(e^{\ln x}=x\) for \(x>0\).
- Both logarithms require positive real arguments.
- Both have domain \((0,\infty)\) and range \((-\infty,\infty)\).
- Common logarithms connect naturally to decimal powers and orders of magnitude.
- Natural logarithms connect naturally to continuous exponential models.
The central distinction is simple: \(\log\) asks for an exponent on \(10\), while \(\ln\) asks for an exponent on \(e\).
Where We Go Next
Common and natural logarithms give us practical ways to evaluate and invert exponential relationships, but more complicated logarithmic expressions require algebraic rules of their own.
The next lesson, Logarithm Laws and Changing Bases, develops the product, quotient, and power laws for logarithms and shows how any logarithmic base can be converted into common or natural logarithms for calculation.