Exponential equations

Exponential functions and logarithms: Logarithmic functions

Exponential equations

We just practiced solving equations with a form similar to #\blue{a}+\log_{\green{b}} \left(x \right)=\purple{c}#. We'll now have a look at equations of the form #\blue{a}^x=\green{b}#.

\[\blue{a}^x=\green{b}\]

gives

\[x=\log_{\blue{a}}\left(\green{b}\right)\]

Example

\[\begin{array}{rcl}\blue{2}^x&=&\green{3}\\x&=&\log_{\blue{2}}\left(\green{3}\right)\end{array}\]

Exact answer
The expression #\log_{\blue{2}}\left(\green{3}\right) = 1.58496250072 \ldots # has infinitely many decimals, and as such cannot be written as a fraction of integers. In cases like these, we'll leave the logarithm in our answer. We call this the exact answer of the equation.

We showed a very simple equation in the above example. However, equations can also be more difficult, as you can see in the examples below.

Solve the equation for #x# :
\[
3^{x+3}=81
\]

Give your answer in the form #x=\ldots# and do not use any exponents in your answer.

#x=1#

\(\begin{array}{rcl}
3^{x+3}&=&81\\
&&\phantom{xxx}\blue{\text{the original equation}}\\
x+3&=&\log_{3}\left(81\right)\\
&&\phantom{xxx}\blue{a^x=b\text{ gives }x=\log_a\left(b\right)}\\
x+3&=&4\\
&&\phantom{xxx}\blue{\text{continue on the logarithm}}\\
x&=&1\\
&&\phantom{xxx}\blue{\text{move constant terms to the right}}\\
\end{array}\)

New example