SOLUTION: log(underscore 3)(x-4) = 2 The book answer tells me the answer is 13 but I have absolutely no idea how to solve it. Thank you for your help!

Question 678737: log(underscore 3)(x-4) = 2
The book answer tells me the answer is 13 but I have absolutely no idea how to solve it. Thank you for your help!
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

Solving equations like this usually starts with transforming the equation into one of the following forms:
log(expression) = other-expression
or
log(expression) = log(other-expression)

Your equation is already in the first form. The next step with this form is to rewrite the equation in exponential form. In general is equivalent to . Using this pattern on our equation we get:

which simplifies to:

Now that the logs are gone we can solve the equation. Adding 4 to each side we get:

When solving logarithmic equations like this you must check your solution(s). It is not optional! You must ensure that all arguments of all logs are positive for each solution. If a "solution" makes any argument of any log zero or negative we must reject that "solution".

Use the original equation to check:

Checking x = 13:

We can already see the argument, (13)-4, will be positive if x = 13. So this solution passes the check! (If 13 had failed this check then we would reject it. And since it was the only "solution" we found rejecting it would mean that the equation has no solutions.)

So the only solution to your equation is x = 13.
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