Logarithmic functions
An exponential function answers the question: what is ${2^5}$? A logarithm answers the reverse question: to what power must we raise ${2}$ to get ${32}$? The answer, ${5}$, is the logarithm of ${32}$ with base ${2}$. This page defines logarithms as the inverses of exponential functions.
Definition of the logarithm
The exponential function ${f(x)=a^x}$ is one-to-one, because its graph always rises (or always falls). So it has an inverse function (see inverse functions). That inverse is the logarithmic function with base ${a}$:
Let ${a>0}$ and ${a\ne 1}$. Then
${\log_a x=y\quad}$ means ${\quad a^y=x}$.
We read ${\log_a x}$ as "log base ${a}$ of ${x}$". In words: ${\log_a x}$ is the exponent you put on ${a}$ to get ${x}$. So a logarithm is an exponent.
Each fact can be written in two forms. The base is the same in both:
| Logarithmic form | Exponential form |
|---|---|
| ${\log_{10}1000=3}$ | ${10^3=1000}$ |
| ${\log_2 \frac{1}{8}=-3}$ | ${2^{-3}=\frac{1}{8}}$ |
| ${\log_5 25=2}$ | ${5^2=25}$ |
| ${\log_9 3=\frac{1}{2}}$ | ${9^{1/2}=3}$ |
Example 1: Write in exponential form:
(a) ${\log_4 64=3}$
(b) ${\log_{10}0.1=-1}$
Write in logarithmic form:
(c) ${3^4=81}$
(d) ${8^{1/3}=2}$
Solution:
(a) ${4^3=64}$
(b) ${10^{-1}=0.1}$
(c) ${\log_3 81=4}$
(d) ${\log_8 2=\dfrac{1}{3}}$
Example 2: Find each value.
(a) ${\log_2 32}$
(b) ${\log_3\dfrac{1}{9}}$
(c) ${\log_4 2}$
(d) ${\log_7 1}$
Solution:
In each case, ask: what power of the base gives the number?
(a) ${2^5=32}$, so ${\log_2 32=5}$.
(b) ${3^{-2}=\dfrac{1}{9}}$, so ${\log_3\dfrac{1}{9}=-2}$.
(c) ${4^{1/2}=\sqrt{4}=2}$, so ${\log_4 2=\dfrac{1}{2}}$.
(d) ${7^0=1}$, so ${\log_7 1=0}$.
Properties of logarithms
These facts follow straight from the definition:
${\log_a 1=0}$, because ${a^0=1}$.
${\log_a a=1}$, because ${a^1=a}$.
${\log_a a^x=x}$ and ${a^{\log_a x}=x}$.
The last line is the inverse function property: ${\log_a}$ undoes ${a^x}$, and ${a^x}$ undoes ${\log_a}$. For example, ${\log_5 5^7=7}$ and ${3^{\log_3 10}=10}$.
Graphs of logarithmic functions
The graph of an inverse function is the reflection across the line ${y=x}$. So the graph of ${y=\log_a x}$ is the graph of ${y=a^x}$ reflected across ${y=x}$. Each point ${(x,y)}$ becomes ${(y,x)}$. For example, ${(2,4)}$ is on ${y=2^x}$, so ${(4,2)}$ is on ${y=\log_2 x}$.
The features of ${a^x}$ also trade places. For ${a>1}$:
The domain of ${\log_a x}$ is ${(0,\infty)}$, and the range is ${(-\infty,\infty)}$.
The graph passes through ${(1,0)}$.
The ${y}$-axis, ${x=0}$, is a vertical asymptote.
Watch out: The logarithm of ${0}$ or of a negative number is undefined. For example, ${\log_2(-4)}$ would be a power of ${2}$ that equals ${-4}$. But every power of ${2}$ is positive.
Example 3: Find the domain of each function.
(a) ${f(x)=\log_3(x-2)}$
(b) ${g(x)=\log_2(6-2x)}$
Sketch the graph of ${f}$.
Solution:
The input of a logarithm must be positive.
(a) ${x-2>0}$, so ${x>2}$. The domain is ${(2,\infty)}$.
(b) ${6-2x>0}$, so ${2x<6}$, and ${x<3}$. The domain is ${(-\infty,3)}$.
The graph of ${f}$ is the graph of ${y=\log_3 x}$ moved ${2}$ units right. Its vertical asymptote moves to ${x=2}$. Two points on it:
${f(3)=\log_3 1=0}$
${f(5)=\log_3 3=1}$
Common and natural logarithms
Two bases are used so often that they have their own names and calculator keys:
The common logarithm has base ${10}$: ${\log x=\log_{10}x}$.
The natural logarithm has base ${e}$: ${\ln x=\log_e x}$.
When no base is written, the base is ${10}$. The natural logarithm ${\ln x}$ is the inverse of ${e^x}$. So ${\ln e^x=x}$ and ${e^{\ln x}=x}$.
Example 4: Find each value. Use a calculator for (e) and (f).
(a) ${\log 100}$
(b) ${\log 0.001}$
(c) ${\ln e^3}$
(d) ${\ln 1}$
(e) ${\log 50}$
(f) ${\ln 10}$
Solution:
(a) ${10^2=100}$, so ${\log 100=2}$.
(b) ${10^{-3}=0.001}$, so ${\log 0.001=-3}$.
(c) ${\ln e^3=3}$, by the inverse property.
(d) ${\ln 1=0}$, because ${e^0=1}$.
(e) ${\log 50\approx 1.699}$. This is between ${1}$ and ${2}$, because ${50}$ is between ${10^1}$ and ${10^2}$.
(f) ${\ln 10\approx 2.303}$.
An application
Logarithms are useful when numbers range from very small to very large. Instead of the number, we use its exponent.
Example 5: Chemists measure how acidic a liquid is with its pH:
${\text{pH}=-\log[\text{H}^+]}$
Here ${[\text{H}^+]}$ is the concentration of hydrogen ions, in moles per liter. Find the pH of
(a) pure water, with ${[\text{H}^+]=10^{-7}}$,
(b) orange juice, with ${[\text{H}^+]=3.2\times 10^{-4}}$.
Solution:
(a) ${\log 10^{-7}=-7}$, so ${\text{pH}=-(-7)=7}$.
(b) With a calculator:
$\text{pH}=-\log(3.2\times 10^{-4})\approx 3.49$
A smaller pH means a more acidic liquid. Each step of ${1}$ in pH means ${10}$ times as many hydrogen ions.
Summary
- ${\log_a x=y}$ means ${a^y=x}$. A logarithm is an exponent.
- ${\log_a 1=0}$, ${\log_a a=1}$, ${\log_a a^x=x}$, and ${a^{\log_a x}=x}$.
- The graph of ${\log_a x}$ is the graph of ${a^x}$ reflected across ${y=x}$. Domain ${(0,\infty)}$; vertical asymptote ${x=0}$; it passes through ${(1,0)}$.
- The logarithm of ${0}$ or of a negative number is undefined.
- Common logarithm: ${\log x=\log_{10}x}$. Natural logarithm: ${\ln x=\log_e x}$.