Advanced Math/Logarithms

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Question
log(base 10)3= a
log(base 10)2= b
Then log(base 5)6=???????
Please answer it.
I am a class 9 student

Answer
Noting that 3 times 2 = 6, write

log10(6) = log10(3・2) = log10(3) + log10(2) = a +b  <-- a and b are known

using the algebraic property of logs.

Also, for the unknown x

log5(6) = x --> 6 = 5^x, and

log10(6) = log10(5^x) = x・log10(5), so that

x = log10(6)/log10(5) = (a+b)/log10(5).

It would be nice to represent log10(5) in terms of a and b, so write

1 = log10(10) = log10(2・5) = log10(2) +log10(5) or

log10(5) = 1 - log10(2) = 1 - b.

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