SOLUTION: I know this shouldn't be hard, we are doing a study sheet for a test and can't get past the second problem. simplify each logarithm (a calculator should not be needed) log ba

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Question 606063: I know this shouldn't be hard, we are doing a study sheet for a test and can't get past the second problem.
simplify each logarithm (a calculator should not be needed)
log base (1/3) 9 so this is 1/3^x equals 9
if I use the change of base formula I get -.5 or 1/2 but when I try to work 1/3^(1/2) I don't get 9
what am I doing wrong, and I can't use a calculator anyway so how do I do this without the change of base formula?
thank you
Answer by jim_thompson5910(35256) (Show Source):
You can put this solution on YOUR website!
Let $x = \log_{\frac{1}{3}}\left(9\right)$

Convert this equation to exponential form to get . Now let's solve for x

Since the bases are equal (to 3), the exponents must be equal. So or

So the solution is

Since we let $x = \log_{\frac{1}{3}}\left(9\right)$ at the top of the problem, we can say that

$\text{\color{red}Answer:} \ \color{blue}\boxed{\log_{\frac{1}{3}}\left(9\right) = \color{red} -2}$

Note: you can use the change of base formula to get the same answer of -2.

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