SOLUTION: Solve the equation log5(x - 7) = 2. Give the solution(s) in fractional form.

Question 371158: Solve the equation log5(x - 7) = 2. Give the solution(s) in fractional form.
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

For equations like yours, where the variable is in the argument of a logarithm, you want the equation in one of the following forms:
log(expression) = other-expression
or
log(expression) = log(other-expression)
You equation, fortunately, is already in the first form. With the first form, the next step is to rewrite the equation in exponential form. In general can be rewritten as . Using this pattern on your equation we get:

which simplifies to:
x-7 = 25
This is a very simple equation to solve for x. Just add 7 to rach side:
x = 32

When solving logarithmic equations it is important, not just a good idea, to check your answer. You must ensure that no arguments (or bases) of any logarithm become zero or negative. And when checking we should use the original equation:

Checking x = 32:

We can already see that the argument of the logarithm is going to be positive. (If 32 had made the argument zero or negative we would have to reject 32 as a solution. And since that was the only "solution" we found, we would end up with no solution!). The rest of the check is nice to do but not as important as making sure the arguments stay positive.

Since , the is indeed 2
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