Solve for X? (Logarithmic Equation)

log_9(x-5) + log_9(x+3) = 1

2 Answers
Apr 17, 2017

x=6 and x = -4

Explanation:

Recall that log_{a} AB = log_{a}A + log_{a} B and log_{a}a = 1.
Therefore, log_{9}(x-5) + log_{9}(x+3) = 1 can be written as log_{9}{(x-5)(x+3)}=log_{9}9.
Cancel out log_{9} on both sides, we have (x-5)(x+3)=9; and solving (x-5)(x+3)-9=0 for x we have x=6, -4

Apr 17, 2017

x=6

Explanation:

Using properties of logarithms we can rewrite the left hand side.

log(a)+log(b)=log(ab)

log_9((x-5)(x+3))=1

Now rewrite both sides in terms of the base 9

9^(log_9((x-5)(x+3)))=9^1

rewriting the left hand side we have

(x-5)(x+3)=9

x^2-2x+15=9

x^2-2x-24=0

(x-6)(x+4)=0

x-6=0 OR x+4=0

x=6

If we are restricted to the real numbers, we disregard
x=-4