Please explain how log(10) 5 can be written in log to the base 6?

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log(10) 5 which is the logarithm to the base 10 of 5 can be written in terms of logarithm to the base 6 in the following way.

Let log(10) 5 = X

=> 5 = 10^X

multiply both sides by 6

=> 6*5 = 6 * 10^X

Take the log to the base 6 of both the sides

=> log(6) [6*5] = log(6) [6*10^X]

Use the relation log a*b = log a + lob

=> log(6) 6 + log(6) 5 = log(6) 6 + log(6) 10^X

We know that log (a) a = 1, or log(6) 6 = 1

=> 1 + log (6) 5 = 1 + X*log(6) 10

=> log (6) 5 = X*log(6) 10

=> X = log(6) 5 / log(6) 10

=> log(10) 5 = log(6) 5 / log(6) 10

We can write log(10) 5 in terms of log to the base 6 as [log(6) 5] / [log(6) 10]

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