Logarithmic Functions

Logarithmic Functions

In mathematics you can find the inverse of an exponential function by switching x and y around: $y=b^{x}\,$ becomes $x=b^{y}\,$. 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: $x=b^{y}\,$ becomes $y=\log _{b}x\,$. If a log is given without a written b then b=10. Also with logarithmic functions, b > 0 and $b\neq 1$. There are 2 cases where the log is equal to x: $\log _{b}b^{X}=X\,$ and $b^{\log _{b}X}=X\,$.

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

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

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

Laws of Logarithmic Functions

When X and Y are positive.

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

Change of Base

When x and b are positive real numbers and are not equal to 1. Then you can write $\log _{a}x\,$ as ${\frac {\log _{b}x}{\log _{b}a}}$. 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 $\log x+\log(x+2)=2$ ($\log$ is another way of writing $\log _{10}$).

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

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

Resources

• Logarithmic Functions, Book Chapter
• Guided Notes