### Exponential functions and logarithms: Exponential functions

### Exponential equations

There is an important rule we can use to solve exponential equations for an unknown variable #x#.

\[\blue{a}^\green{b}=\blue{a}^\purple{c}\]

gives

\[\green{b}=\purple{c}\]

**Example**

\[\begin{array}{rcl}\blue{3}^\green{x}&=&9\\\blue{3}^\green{x}&=&\blue{3}^\purple{2}\\ \green{x}&=&\purple{2}\end{array}\]

Solve the equation for #x# :

\[

4^{x+1}=256

\]

Do not use exponents and give your final answer in the form #x=\ldots#.

Simplify the number at the dots as much as possible.

#x=3#

\(\begin{array}{rcl}

4^{x+1}&=&256\\

&&\blue{\text{the original equation}}\\

4^{x+1}&=&4^4\\

&&\blue{\text{write \(256\) as a power of \(4\)}}\\

x+1&=&4\\

&&\blue{a^b=a^c\text{ gives }b=c}\\

x&=&3\\

&&\blue{\text{constant terms moved to the right}}\\

\end{array}\)

\(\begin{array}{rcl}

4^{x+1}&=&256\\

&&\blue{\text{the original equation}}\\

4^{x+1}&=&4^4\\

&&\blue{\text{write \(256\) as a power of \(4\)}}\\

x+1&=&4\\

&&\blue{a^b=a^c\text{ gives }b=c}\\

x&=&3\\

&&\blue{\text{constant terms moved to the right}}\\

\end{array}\)

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