Final answer:
To solve the equation log₅(x+12) + log₅(x-12) = 2, combine the log terms into log₅((x+12)(x-12)) = 2 using the product rule for logarithms, then convert to exponential form and solve for x, resulting in x = 13 after checking for extraneous solutions.
Step-by-step explanation:
To solve the equation log₅(x+12) + log₅(x-12) = 2, we must first use the properties of logarithms to combine the two log terms. According to the product rule for logarithms, which states that logₙ(a) + logₙ(b) = logₙ(ab), we can combine them into a single logarithm:
log₅((x+12)(x-12)) = 2
This simplifies to:
log₅(x² - 144) = 2
Now, to remove the logarithm, we rewrite the equation in exponential form:
5² = x² - 144
This gives us:
25 = x² - 144
Now, add 144 to both sides to isolate the x² term:
x² = 169
Finally, take the square root of both sides, remembering to consider both the positive and negative roots:
x = ±√169
x = ± 13
However, since the original equation contains log terms, we must check for extraneous solutions. We can only take the logarithm of positive numbers, so x must be greater than 12 to be a valid solution since x-12 must be positive. Thus, the only solution is x = 13.