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)}