SOLUTION: How do you solve a problem that has a divided variable as a base of a logarithm? My problem is Loga(x)/Log a/b(x)

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Question 461940: How do you solve a problem that has a divided variable as a base of a logarithm?
My problem is Loga(x)/Log a/b(x)

Answer by AnlytcPhil(1806) About Me  (Show Source):
You can put this solution on YOUR website!
log%28a%2C%28x%29%29%2Flog%28a%2Fb%2C%28x%29%29
Using the change of base formula on the numerator only to
natural logs:

NUMERATOR+=log%28a%2C%28x%29%29=ln%28x%29%2Fln%28a%29

Using the change of base formula on the denominator only to
natural logs: [Note: you could use logs base 10 if you prefer]

DENOMINATOR+=+log%28a%2Fb%2C%28x%29%29=ln%28x%29%2Fln%28a%2Fb%29

So we have

log%28a%2C%28x%29%29%2Flog%28a%2Fb%2C%28x%29%29%22%22=%22%22NUMERATOR%22%F7%22DENOMINATOR%22%22=%22%22ln%28x%29%2Fln%28a%29%22%F7%22ln%28x%29%2Fln%28a%2Fb%29%22%22=%22%22ln%28x%29%2Fln%28a%29%22%22%2A%22%22ln%28a%2Fb%29%2Fln%28x%29%22%22=%22%22cross%28ln%28x%29%29%2Fln%28a%29%22%22%2A%22%22ln%28a%2Fb%29%2Fcross%28ln%28x%29%29%22%22=%22%22ln%28a%2Fb%29%2Fln%28a%29%22%22=%22%22%28ln%28a%29+-+ln%28b%29%29%2Fln%28a%29%22%22=%22%22ln%28a%29%2Fln%28a%29-ln%28b%29%2Fln%28a%29+=+1-ln%28b%29%2Fln%28a%29

If you prefer to use logs base 10 then just replace 
all the %22ln%28_%29%22's with %22log%28_%29%22's and the answer will be
1-%22log%28b%29%22%2F%22log%28a%29%22.

Edwin