Logarithmic equations

Exponential functions and logarithms: Logarithmic functions

Logarithmic equations

We call equations of the form #\log_{\blue{a}}\left(x\right)=\green{y}# logarithmic equations. We can use the rule explained below to solve equations like this.

\[\log_{\blue{a}}\left(x\right)=\green{y}\quad \text{gives}\quad x=\blue{a}^\green{y}\]

Example

\[\begin{array}{rcl}\log_{\blue{2}}\left(x\right)&=&\green{4}\\x&=&\blue{2}^{\green{4}}\end{array}\]

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

Solve for #x# :
\[\log_{5}\left(x-16\right)=3.\]

Give your answer in the form #x=\ldots# and simplify as much as possible.

#x=141#

\(\begin{array}{rcl}
\log_{5}\left(x-16\right)&=&3\\
&&\phantom{xxx}\blue{\text{the original equation}}\\
x-16&=&125\\
&&\phantom{xxx}\blue{\log_{a}\left(x\right)=b\text{ gives }x=a^b}\\
x&=&141\\
&&\phantom{xxx}\blue{\text{moved the constant terms to the right}}\\
\end{array}\)

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