If log23=a, log35=b, log72=c, then the logarithm of the number 63 to base 140 is

To solve this problem, we need to express 63 and 140 in terms of the given bases 2, 3, 5, and 7.

First, let's express 63 and 140 in terms of 2, 3, 5, and 7:

63 = 3^2 * 7 140 = 2^2 * 5 * 7

Now, let's express the logarithm of 63 to base 140 using the change of base formula:

log_140(63) = log(63) / log(140)

We can express the numerator and denominator in terms of the given bases:

log(63) = log(3^2 * 7) = 2log(3) + log(7) = 2a + c log(140) = log(2^2 * 5 * 7) = 2log(2) + log(5) + log(7) = 2a + b + c

Substituting these into the change of base formula gives:

log_140(63) = (2a + c) / (2a + b + c)

This is the logarithm of the number 63 to base 140 in terms of the given variables a, b, and c.