Question

Logs equal logs?

`\log_mx=\log_nx`

Answer 1

Yes

Explanation:

We are basically trying to prove that `m=n` if `log_mx = log_nx` for all `x > 0`. We can prove this with a proof by contradiction.

Let `y = log_mx = log_nx`. Assume `n ne m`.

By definition,
`m^y = m^(log_mx) = x` and `n^y = n^(log_nx) = x`
so we know that
`m^x = n^x`

Now since `x` is a positive real, we can take the `xth` root of both sides without problem, yielding `m=n`, violating our initial assumption and proving the statement above.