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I stumbled across this weird graph while working with logarithms, specifically I found that for $x\geq0$, $4^{\log_2 x}=x^2$. I've been trying to understand why this is the case for a while now, but can't find a generalization of this phenomenon. I know it probably has something to do with the logarithmic identity that $a^{\log_a x}=x$, but I don't see how since in this case, our bases differ by a factor of $2$. Why does this equality work, and is there a general form of it beyond exponent base=4 and log base=2?

moistnar
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1 Answers1

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$$ 4^{\log_2 x}=(2^2)^{\log_2 x}=2^{2\log_2 x}=2^{\log_2 x^2}=x^2$$

using the logarithmic identity along with the law of exponentiation $(a^b)^c=a^{bc}$ and the law of logarithms $k\log_b x=\log_b x^k$.

We could say in general:

$$ (b^k)^{\log_b x}=x^k $$

A.M.
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