Question

How can a logarithmic equation be solved by graphing?

Answer 1

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 (x - 1)`

This can be converted into a linear equation by understanding that `a = log_b n -> b^a = n`.

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

`{(y_1 = x - 1), (y_2 = 4):}`

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

Here is yet another example:

Solve the equation `4 = log_2 (x + 3) + log_2 (4x)`

This can be written as a single logarithm:

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

`4 = log_2 (4x^2 + 12x)`

Rewrite without logarithms:

`16 = 4x^2 + 12x`

Graph the two equations:

`{(y_1 = 4x^2 + 12x), (y_2 = 16):}`

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!