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My teacher told me in order to undo $\log$ you have to base both sides with the same number of the existing $\log$, but I don’t really understand why 4 and the power which is $\log_4$ cancels each other out, someone please tell me the process of it, not just the formula which I already know.

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4 Answers

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Note that by definition

$$x=\log_a b \iff a^x=b$$

then

$$a^{\log_a b}=b$$

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$a^{\log_a b}=b$ and $\log_a {a^b}=b$, by the very definition...

These are inverse functions...

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You're basically asking simplification of $$a^{\log_a b} \tag 1$$

So suppose $$\log_b a =c$$

By definition of log, we've $$\log_b a =c \implies a^c =b \tag2$$

Now using $(2)$ and $(1)$, we get $$a^{\log_a b}=a^{(c)}\tag3 $$

Now using $(3)$ and $(2)$, finally we've $$a^{\log_a b} =b$$

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I think your confusion is in the statement that the 4 and the $\log_4$ "cancel each other". That's not very precise but it's a simplification of what's really happening. Think of $4^a$ as starting at $a$ and taking four steps forward and $\log_4 a$ taking four steps back. Then $4^{\log_4 a}$ would just put you back at $a$. You can read this is "Take four to the power of the number that you need to raise 4 to in order to get $a$."

So in your example, $4^{\log_4 64}$ would be read "Take four to the power of the number that you need to raise 4 to in order to get 64." The number you need to raise 4 to in order to get 64 is 3, so take 4 raised to that power and you get $4^3 = 64$.

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