Why can we take the log of both sides?

Question

I was watching a video that proves the "Log of a power" rule.

1. I'm just having trouble understanding the loga(a^x) = x rule - which he uses in the proof

2. And I don't get why you can log both sides. I know whatever you do to one side of a equation you can do to the other - but I still think there's more to it than just that shallow understanding. As soon as I log something I am saying its a exponent - I am basically going from working with exponentiations to working with exponents - what are the steps behind this?

3. Also if 10^x = 10^2 - the bases would be the same - so intuitively I would say that the only differences can possibly be in the exponents - so x = 2. But what is the actual way to prove this? Can you show me all the steps that get us to x = 2

Answer 1

Here use that the function $f(x)=\log_a(x)$ is well defined, that is $a=b \Rightarrow f(a) = f(b)$. Then $$a^{mn} = x^n \Rightarrow \log_a(a^{nm})= \log_a(x^n)$$

And the facts $\log_aa^{mn} = mn\log_aa$, $\log_a a = 1$ and $\log_ax^n = n\log_a x$.

Answer 2

As Hakim points out in the comments, doing an operation to both sides of an equation is logically valid because logs are well-defined functions.

If you still don't like the given proof, then here's an equivalent proof that does everything in a single chain of equalities but uses substitutions instead: \begin{align*} \log_a(x^n) &= \log_a((a^{\log_a x})^n) &\text{by substituting } x = a^{\log_a x}\\ &= \log_a(a^{n\log_a x}) &\text{since in general $(b^c)^d = b^{cd} = b^{dc}$}\\ &= n\log_a x &\text{since in general} \log_m (m^y) = y \end{align*}