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.
Logarithmic equations
\[\log_{\blue{a}}\left(x\right)=\green{y}\quad \text{gives}\quad x=\blue{a}^\green{y}\]
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.
#x=36#
\(\begin{array}{rcl}
\log_{3}\left(x-9\right)&=&3\\
&&\phantom{xxx}\blue{\text{the original equation}}\\
x-9&=&27\\
&&\phantom{xxx}\blue{\log_{a}\left(x\right)=b\text{ gives }x=a^b}\\
x&=&36\\
&&\phantom{xxx}\blue{\text{moved the constant terms to the right}}\\
\end{array}\)
\(\begin{array}{rcl}
\log_{3}\left(x-9\right)&=&3\\
&&\phantom{xxx}\blue{\text{the original equation}}\\
x-9&=&27\\
&&\phantom{xxx}\blue{\log_{a}\left(x\right)=b\text{ gives }x=a^b}\\
x&=&36\\
&&\phantom{xxx}\blue{\text{moved the constant terms to the right}}\\
\end{array}\)
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