Question

How do you convert a logarithm to a different base?

Answer 1

If you want to convert `log_a(x)` into `log_b(x)`, use the following equality:
`log_b(x) = log_b(a)*log_a(x)`

Explanation:

Consider `y=log_a(x)`.
From a definition of logarithm
(1) `x = a^y`
Analogously, if `z=log_b(x)`,
(2) `x = b^z`
Finally, if `r=log_b(a)`,
(3) `a = b^r`

From (1) and (2) we derive
(4) `a^y = b^z`

Using (3) and substituting `a` in (4) we get
(5) `(b^r)^y = b^z`
or, simplifying the left part,
(6) `b^(r*y) = b^z`

From the last equation we see that
`z = r*y`
or, returning back to the meaning of `r`, `y` and `z`,
`log_b(x)=log_b(a)*log_a(x)`