SOLUTION: Solve the following: log(z^2-25)-log(z+5)=log7

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Question 114805: Solve the following:
log(z^2-25)-log(z+5)=log7
Answer by solver91311(6137) About Me  (Show Source):
You can put this solution on YOUR website!
log%28%28z%5E2-25%29%29-log%28%28z%2B5%29%29=log7

First notice that z%5E2-25 is the difference of two squares, so log%28%28z%5E2-25%29%29=log%28%28%28z%2B5%29%28z-5%29%29%29. But we also know that the log of the product is the sum of the logs of the factors (log%28%28ab%29%29=log%28a%29%2Blog%28b%29), so we can write:

log%28%28z%2B5%29%29%2Blog%28%28z-5%29%29-log%28%28z%2B5%29%29=log7

Since log%28%28z%2B5%29%29-log%28%28z%2B5%29%29=0, our equation reduces to:

log%28%28z-5%29%29=log7

Since log%28a%29=log%28b%29 if and only if a=b, we can now write:

z-5=7
z=12 Done.