SOLUTION: use the change of base formula to evaluate log6 12. then convert log6 12 to a logarithm in base 3. round to the nearest thousandth

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Question 957905: use the change of base formula to evaluate log6 12. then convert log6 12 to a logarithm in base 3. round to the nearest thousandth
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Since 12 is not a well-known power of 6, we will need to use a calculator to evaluate log%286%2C+%2812%29%29. Since most calculators do not have a button for base 6 logs we will need to use the change of base formula to convert to logarithms of a base our calcuator does have. If calculators have any buttons for logarithms at all they have buttons for base 10 and/or base e ("log" and/or "ln"). So we will convert log%286%2C+%2812%29%29 to one of these bases.

The change of base formula is log%28a%2C+%28p%29%29+=+log%28b%2C+%28p%29%29%2Flog%28b%2C+a%29%29. We can use this to change a base 6 log to base 10:
log%286%2C+%2812%29%29+=+log%28%2812%29%29%2Flog%28%286%29%29
... or to base e logs:
log%286%2C+%2812%29%29+=+ln%2812%29%2Fln%286%29

I will leave it up to you and your calculator to find the decimal value. (Just be sure to find the two logs first, then divide them.) The answer is the same whether you use log or ln. (Try it and see!)

To change log%286%2C+%2812%29%29 into base 3 logs:
log%286%2C+%2812%29%29+=+log%283%2C+%2812%29%29%2Flog%283%2C+%286%29%29