We want to prove the identity `log_(1/p)(x) = -log_p(x)` (for all `xgt0,` `pgt0` and `p != 1` ).
To do this, let's recall what logarithms are. By the definition, `log_b(a)` is such a number `c` that `b^c = a.` It is known that such a number always exists and is unique (of course for `agt0,` `bgt0` and `b != 1` ).
Therefore to verify this identity we raise `1/p` to the power of each side:
`(1/p)^(log_(1/p)(x)) = (1/p)^(-log_p(x))`
(the function `(1/p)^x` is one-to one on its domain, therefore this operation gives an equivalent equality).
The left part is equal to `x` by the definition of logarithm, what about the right part? We'll use some properties of powers, `1/p=p^(-1)` and `(p^a)^b=p^(a*b):`
`(1/p)^(-log_p(x)) =(p^(-1))^(-log_p(x)) =p^((-1)*(-log_p(x))) =p^(log_p(x)).`
And this is equal to `x,` too, by the definition of logarithm. This way we proved the desired identity.