Question

How do you solve `log_2 (3x-9) - 2 = log_2 (x-4)`?

Answer 1

Get in the form log(A)=B then re-write in exponential form.

Explanation:

Get all logs on one side, then use quotient rule for logarithms `log A - log B = log(A/B)` to combine expressions:
`log_2(3x-9)-log_2(x-4)=2`
`log_2((3x-9)/(x-4))`=2

We know that `log_2(4)=2`, so...

`(3x-9)/(x-4) = 4`

Multiply both sides by x-4, subtract 3x and add 16 to both sides.

x=7.

IMPORTANT: When solving log equations, check for extraneous solutions: Log of 0 or less is undefined for real numbers so make sure that substituting your value for x will not give you a 0 or negative inside the log function. In this case:

`log_2(12)-2=log_2(3)`

so our solution is ok.