### 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=29#

\(\begin{array}{rcl}

\log_{3}\left(x-20\right)&=&2\\

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

x-20&=&9\\

&&\phantom{xxx}\blue{\log_{a}\left(x\right)=b\text{ gives }x=a^b}\\

x&=&29\\

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

\end{array}\)

\(\begin{array}{rcl}

\log_{3}\left(x-20\right)&=&2\\

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

x-20&=&9\\

&&\phantom{xxx}\blue{\log_{a}\left(x\right)=b\text{ gives }x=a^b}\\

x&=&29\\

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

\end{array}\)

Unlock full access

Teacher access

Request a demo account. We will help you get started with our digital learning environment.

Student access

Is your university not a partner?
Get access to our courses via

Or visit omptest.org if jou are taking an OMPT exam.

**Pass Your Math**independent of your university. See pricing and more.Or visit omptest.org if jou are taking an OMPT exam.