Solve the equation log3 (x) + log3 (x − 8) = 2

Question

The equation is log₃(x) + log₃(x − 8) = 2. Sorry for any confusion. I am trying to solve this problem and I have gotten this far into solving it:

log₃(x(x-8))=2

log_3(X2_-8x)=2

_{but I'm not sure how to continue after that. Can anyone help me?}

Answer 1

Hi Ariel, you are close to the answer. Remember y = b^x is the same as log_b y = x

Rewrite log₃ (x² - 8x) = 2 as 3² = x² - 8x

This becomes:

x² - 8x = 3², or 9

Rewrite the equation as:

x² - 8x - 9 = 0

Factor:

(x - 9) (x + 1) = 0

Use the zero product rule:

x - 9 = 0

x = 9

And

x + 1 = 0

x = -1

Now test each answer by substituting into log₃ (x² - 8x) = 2

First, x = 9:

log₃ (9² - 8 * 9) = 2

log₃ (81 - 72) = 2

log₃ 9 = 2, which is true

Second, x = -1:

log₃ ((-1)² - 8 * (-1)) = 2

log₃ (1 + 8) = 2

log₃ 9 = 2, which is true

Answer: x = 9 and x = -1

Questions?