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# :
\[
5^{x+1}=25
\]
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=1#
\(\begin{array}{rcl}
5^{x+1}&=&25\\
&&\blue{\text{the original equation}}\\
5^{x+1}&=&5^2\\
&&\blue{\text{write \(25\) as a power of \(5\)}}\\
x+1&=&2\\
&&\blue{a^b=a^c\text{ gives }b=c}\\
x&=&1\\
&&\blue{\text{constant terms moved to the right}}\\
\end{array}\)
\(\begin{array}{rcl}
5^{x+1}&=&25\\
&&\blue{\text{the original equation}}\\
5^{x+1}&=&5^2\\
&&\blue{\text{write \(25\) as a power of \(5\)}}\\
x+1&=&2\\
&&\blue{a^b=a^c\text{ gives }b=c}\\
x&=&1\\
&&\blue{\text{constant terms moved to the right}}\\
\end{array}\)
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