8. Solve for
x log_{4} (x + 3) - log_{4} (x-1) = 1

First express the left side as a single log.

Now change to exponential form

Clear denominators

Remove parentheses

Transpose

Combine

Divide

Check:

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

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

Each term is the log of a quotient

Remove parentheses

The logs of 3 cancel

16 and 4 are both powers of 4 so this becomes

which checks.