How can a logarithmic equation be solved by graphing?

1 Answer
Noah G
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 (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!

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