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:
$\endgroup$ 2$(1)$ 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 $\log_ax^p=p\cdot\log x$
1 Answer
$\begingroup$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
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