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.

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

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