[Math] Why can we take the log of both sides

logarithms

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

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Best Answer

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 $\log_ax^n = n\log_a x$.

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