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Can someone explain how to get from steps $(1)$ to $(2)$ in this proof of the logarithm power rule? How is the "equivalent exponential expression found? The power rule:

For any positive $x$, any number $p$, and $a>0,a\ne1$, $$\log_ax^p=p\cdot\log_ax$$

The proof:

$(1)$ Let $b=\log_ax$.

$(2)$ Then, writing an equivalent exponential equation, we have $x=a^b$.

$(3)$ We raise both sides of the latter equation to the $p$th power, getting $x^p=a^{bp}$

$(4)$ Now we can write an equivalent logarithmic equation: $\log_ax^p=\log_aa^{pb}$, which simplifies to $\log_ax^p=pb$.

$(5)$ But $b=\log_ax$, so we have $\log_ax^p=p\cdot\log x$

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    Use definition of $\log_a(x)$ as that number to which $a$ must be raised to get $x.$ – 2017-02-23
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    See whether this helps you: https://en.m.wikipedia.org/wiki/List_of_logarithmic_identities#Cancelling_exponentials – 2017-02-23

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That's basically just using the definition of a logarithm.

The statement $x = a^b$ is equivalent to $a^{log_ax} = a^b$. We're raising $a$ to the power of both sides.

Just like $e$ and the natural log $\ln$ or $\log_e$, $a$ and $\log_a$ cancel out