SOLUTION: log base of 9 (x^2+12x+17)=2

Question 274273: log base of 9 (x^2+12x+17)=2
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

To solve a logarithmic equation where the variable is in the base or the argument of a logarithm, you start by getting the equation into one of the following forms:
log(variable-expression) = other-expression
or
log(variable-expression) = log(other-expression)

Fortunately your equation is already in the first form. With this form we proceed by rewriting the equation in exponential form:

which simplifies to:

Now we solve this equation. Since it is a quadratic equation, we'll get one side equal to zero (by subtracting 81 from each side):

Then we factor it (or use the Quadratic Formula). This factors fairly easily:

From the Zero Product Property we know that this product is zero only if one of the factors is zero. So:
or
Solving these we get:
or

When solving logarithmic equations it is important (not just a good idea) to check your answers. We must make sure that no bases or arguments of logarithms are zero or negative. Always use the original equation to check:

Checking x = -16:

which simplifies to:

And since , Check!

Checking x = 4:

which simplifies to:

And since , Check!

Both solutions check out. If either or both solutions had made the base or the argument of a logarithm zero or negative we would have to reject it/them.
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