# Logarithmic Functions

## Logarithmic Functions

In mathematics you can find the inverse of an exponential function by switching x and y around: ${\displaystyle y=b^{x}\,}$ becomes ${\displaystyle x=b^{y}\,}$. The problem arises on how to find the value of y. The logarithmic function solved this problem. All conversions of logarithmic function into an exponential function follow the same pattern: ${\displaystyle x=b^{y}\,}$ becomes ${\displaystyle y=\log _{b}x\,}$. If a log is given without a written b then b=10. Also with logarithmic functions, b > 0 and ${\displaystyle b\neq 1}$. There are 2 cases where the log is equal to x: ${\displaystyle \log _{b}b^{X}=X\,}$ and ${\displaystyle b^{\log _{b}X}=X\,}$.

To recap, a logarithm is the inverse function of an exponent.

e.g. The inverse of the function ${\displaystyle f(x)=3^{x}}$ is ${\displaystyle f^{-1}(x)=\log _{3}x}$.

In general, ${\displaystyle y=b^{x}\iff x=\log _{b}y}$, given that ${\displaystyle b>0}$.

### Laws of Logarithmic Functions

When X and Y are positive.

• ${\displaystyle \log _{b}XY=\log _{b}X+\log _{b}Y\,}$
• ${\displaystyle \log _{b}{\frac {X}{Y}}=\log _{b}X-\log _{b}Y\,}$
• ${\displaystyle \log _{b}X^{k}=k\log _{b}X\,}$

### Change of Base

When x and b are positive real numbers and are not equal to 1. Then you can write ${\displaystyle \log _{a}x\,}$ as ${\displaystyle {\frac {\log _{b}x}{\log _{b}a}}}$. This works for the natural log as well. here is an example:

### Solving a Logarithmic Equation

A logarithmic equation is an equation wherein one or more of the terms is a logarithm.

e.g. Solve ${\displaystyle \log x+\log(x+2)=2}$ (${\displaystyle \log }$ is another way of writing ${\displaystyle \log _{10}}$).

{\displaystyle {\begin{aligned}\log x+\log(x+2)&=2\\\log(x(x+2))&=2\\x(x+2)&=100\\x^{2}+2x&=100\\(x+1)^{2}&=101\\x+1&={\sqrt {101}}\\x&=-1\pm {\sqrt {101}}\end{aligned}}}

${\displaystyle \log _{2}8={\frac {\log 8}{\log 2}}={\frac {.9}{.3}}=3\,}$ now check ${\displaystyle 2^{3}=8\,}$