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log(base 10)3= a

log(base 10)2= b

Then log(base 5)6=???????

Please answer it.

I am of class 9

My computer has been tied up, but finally answering your question ...

I will used [N] to designate an equation that is used later.

[1] If is known that log(base b)a = ln(a)/ln(b).

Using [1] on the 1st line of your question gives log(base 10)3 = ln(3)/ln(10) = a, so

[2] ln(3) = a*ln(10).

Using [1] on the 2nd line gives log(base 10)2 = ln(2)/ln(10) = b, so

[3] ln(2) = b*ln(10).

Using [1] on the question in the 3rd line of the question gives

[4] log(base 5)6 = ln(6)/ln(5).

Another property of log()'s is

[5] log(ab) = log(a) + log(b). Using [5], we know that

[6] ln(6) = ln(2) + ln(3).

Putting [6] into [4] gives

[7] log(base 5)6 = {ln(2) + ln(3)}/ln(5).

Using [7] with [2] and [3] gives [8] log(base 5)6 = {b*ln(10) + a*ln(10)}/ln(5).

Now for [8], we can factor out a ln(10) and get the answer,

log(base 5)6 = (b+a)ln(10)/ln(5).

Since log(b)/log(a) is the same no matter what the base is,

this can also be said to be (b+a)log(10)/log(5).

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