## Logarithmic and Exponential Laws

These are important to know both for when we are graphing these functions and algebraically solving these equations.

## Exponential Equations

First we will look at exponential equations and the first rule.

$a^{x}=a^{y} \rightarrow x=y$

### Index Laws

Here are a few of index laws, which are used to solve exponential equations:

• $a^{m}\times a^{n}=a^{m+n}$
• $a^{m}\div a^{n}=a^{m-n}$
• $(a^{m})^{n}=a^{mn}$
• $a^{0}=1$

### Exponential Example

2013 exam 1 Question 5 b

Solve $3^{-4x}=9^{6-x}$, for x

Our first step is to make the base numbers the same

$3^{-4x}=3^{2(6-x)}$

Then we can equate the powers and solve for x.

$-4x=2(6-x) \newline -2x=12 \newline x=-6$

## Logarithmic Functions

The log function is simply the inverse of the the exponential function, it is used to model functions that grow exponentially and also to transform equations.

$log_{e}(2^{3}) \rightarrow 3log_{e}(2)$

Also if we have a log of a number the same as the base then it equals 1.

$log_{e}(e)=1$

$log_{e}(e^{3}) \rightarrow 3log_{e}(e) \rightarrow 3$

### Logarithmic Laws

There are a few logarithmic laws ,which will help us to perform our algebraic operations.

• $log_{a}(m)+log_{a}(n)=log_{a}(mn)$
• $log_{a}(m)-log_{a}(n)=log_{a}(\frac{m}{n})$
• $log_{a}(\frac{1}{n}) \rightarrow log_{a}(n^{-1})\rightarrow -log_{a}(n)$

### Logarithmic Example

For an example lets take an example of question 6 from the 2014 exam one.

Solve: $log_{e}(x)-3=log_{e}(\sqrt{x}), for \geq 0$

Our first step is to group the logs onto one side.

$log_{e}(x)-log_{e}(\sqrt{x})=3 \newline log_{e}(\frac{x}{\sqrt{x}})=3 \newline \frac{x}{\sqrt{x}}=e^{3} \newline \sqrt{x}=e^{3} \newline x=e^{6}$