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 $x$ :
$\log_{5}\left(x-2\right)=3.$

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

$x=127$

$\begin{array}{rcl} \log_{5}\left(x-2\right)&=&3\\ &&\phantom{xxx}\blue{\text{the original equation}}\\ x-2&=&125\\ &&\phantom{xxx}\blue{\log_{a}\left(x\right)=b\text{ gives }x=a^b}\\ x&=&127\\ &&\phantom{xxx}\blue{\text{moved the constant terms to the right}}\\ \end{array}$

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