8. Solve for x log4 (x + 3) - log4 (x-1) = 1

First express the left side as a single log.

Now change to exponential form

Clear denominators

x + 3 = 4(x - 1)

Remove parentheses

x + 3 = 4x - 4

Transpose

4 + 3 = 4x - x

Combine

7 = 3x

Divide

x = 7/3

Check:

Copy down the original equation except that wherever you see an x, copy down the solution in parentheses.

log4((7/3) + 3) - log4((7/3) - 1) = 1

Find common denominators to add and subtract the fractions inside the parentheses.

log4(7/3 + 9/3) - log4(7/3 - 3/3) = 1

log4(16/3) - log4(4/3) = 1

Each term is the log of a quotient

(log416 - log4 3) - (log44 - log4 3) = 1

Remove parentheses

log416 - log4 3 - log44 + log4 3 = 1

The logs of 3 cancel

log416 - log44 = 1

16 and 4 are both powers of 4 so this becomes

2 - 1 = 1

which checks.