## Summary

IDefinition, domain, and rangeIIGraphical representationIIIProperties of logarithmic functions## Definition, domain, and range

Suppose a is a positive real number not equal to 1 and f\left(x\right)=a^x is the exponential function with base a. The graph of f\left(x\right) passes the horizontal line test. Therefore f\left(x\right) has an inverse function.

### Logarithmic function

Let a be a positive real number not equal to 1. The logarithmic function with base a is the inverse of the exponential function a^x and is denoted as follows:

** \log_a\left(x\right) **

In particular, if b and c are real numbers then we have:

** \log_a\left(b\right)=c if and only if b=a^c **

Consider the following logarithmic function:

f\left(x\right)=\log_3\left(x\right).

We know that:

3^2=9

Therefore:

\log_3\left(9\right)=2

Consider the following logarithmic function:

f\left(x\right)=\log_{10}\left(x\right).

We know that:

10^3=1\ 000

Therefore:

\log_{10}\left(1\ 000\right)=3

More generally, \log_a\left(b\right) is equal to the power one would need to raise a to in order to obtain b.

We know that:

7^2=49

Therefore:

\log_7\left(49\right)=2

The natural logarithm is the function \ln\left(x\right) which is the logarithm function whose base is the number e.

Consider the natural logarithm function:

f\left(x\right)=\ln\left(x\right)

Then we have the following:

\ln\left(e^2\right)=2

If a is a positive number not equal to 1 then the domain of a^x is all real numbers, \mathbb{R}, and the range of a^x is \left(0,\infty\right). Therefore:

- The domain of \log_a\left(x\right) is \left(0,\infty\right).
- The range of \log_a\left(x\right) is all real numbers, \mathbb{R}.

## Graphical representation

Let a be a positive real number not equal to 1. Because \log_a\left(x\right) is the inverse function of a^x the graph of \log_a\left(x\right) is obtained by reflecting the graph of a^x across the line y=x.

Let a \gt 0 be a positive real number not equal to 1 and consider the logarithmic function f\left(x\right)=\log_a\left(x\right).

Then by the above we have the following:

- The graph of \log_a\left(x\right) increases from left to right if a \gt 0.
- The graph of \log_a\left(x\right) decreases from left to right if 0 \lt a \lt 1

- The graph of \log_2\left(x\right) increases from left to right.
- The graph of \log_{\frac{1}{2}}\left(x\right) decreases from left to right.

## Properties of logarithmic functions

### Product rule for logarithms

Let a be a positive number not equal to 1. Then for any positive real numbers x and y :

** \log_a\left(xy\right)=\log_a\left(x\right)+\log_a\left(y\right) **

Consider the following logarithmic function:

\log_{10}\left(x\right)

We have the following:

\log_{10}\left(1\ 200\right)\\=\log_{10}\left(12\cdot 100\right)\\=\log_{10}\left(12\right)+\log_{10}\left(100\right)\\=\log_{10}\left(12\right)+2

### Division rule for logarithms

Let a be a positive number not equal to 1. Then for any positive real numbers x and y :

** \log_a\left(\dfrac{x}{y}\right)=\log_a\left(x\right)-\log_a\left(y\right) **

Consider the following logarithmic function:

f\left(x\right)=\log_2\left(x\right)

Then we have the following:

\log_2\left(\dfrac{2}{x}\right)=\log_2\left(2\right)-\log_2\left(x\right)\\=1-\log_2\left(x\right)

### Power rule for logarithms

Let a be a positive number not equal to 1. Then for any positive real number x and any real number y :

** \log_a\left(x^y\right)=y\log_a\left(x\right) **

Consider the following logarithmic function:

f\left(x\right)=\log_7\left(x\right)

Then we have the following:

\log_7\left(8\right)=\log_7\left(2^3\right)=3\log_7\left(2\right)

### Change of base

Let a and b be positive real numbers, neither of which are 0. Then for any positive real number x :

** \log_a\left(x\right)=\dfrac{\log_b\left(x\right)}{\log_b\left(a\right)} **

Consider the following logarithmic function:

f\left(x\right)=\log_2\left(x\right)

Then we have the following:

\log_{2}\left(10\right)=\dfrac{\log_{10}\left(10\right)}{\log_{10}\left(2\right)}=\dfrac{1}{\log_{10}\left(2\right)}