# How can a logarithmic equation be solved by graphing?

Jan 17, 2017

There are a couple of steps.

a) Separate into functions and graph

b) Locate the intersection points.

Here is an example.

Solve the equation $2 = {\log}_{2} \left(x - 1\right)$

This can be converted into a linear equation by understanding that $a = {\log}_{b} n \to {b}^{a} = n$.

So, $4 = x - 1$. Here, it is obvious that $x = 5$, but if we have to solve graphically, we separate as:

$\left\{\begin{matrix}{y}_{1} = x - 1 \\ {y}_{2} = 4\end{matrix}\right.$

Graph both lines and locate the intersection point, which is $x = 5$.

Here is yet another example:

Solve the equation $4 = {\log}_{2} \left(x + 3\right) + {\log}_{2} \left(4 x\right)$

This can be written as a single logarithm:

4 = log_2((x + 3)4x))

$4 = {\log}_{2} \left(4 {x}^{2} + 12 x\right)$

Rewrite without logarithms:

$16 = 4 {x}^{2} + 12 x$

Graph the two equations:

$\left\{\begin{matrix}{y}_{1} = 4 {x}^{2} + 12 x \\ {y}_{2} = 16\end{matrix}\right.$

You will find the intersection point is $x = 1$ and $x = - 4$. The $x = - 4$ is extraneous though, due to the domain of the logarithmic function. This is why it is vital to check our solutions algebraically.

Hopefully this helps!