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I need to prove the logarithmic power rule: $$\log_b(x^r) = r \cdot \log_b(x) \hspace{2.3cm}$$

I have seen a large number of sources citing a similar proof, which goes like this:

\begin{align*} \text{let} \ \ m &= \log_b(x), \\ x &= b^m \\ x^r &= (b^m)^r \\ \log_b(x^r) &= \log_b((b^m)^r) \\ &= \log_b(b^{mr})& \text{(5)}\\ &= mr & \text{(6)}\\ &= rm \\ \log_b(x^r) &= r \cdot \log_b(x) & \text{Q.E.D} \\ \end{align*}

The problem I have with this proof is that in going from steps 5 to 6, I implicitly justify them as $$\hspace{0.7cm} \log_b(b^{mr}) = mr \cdot \log_b(b) = mr \cdot 1 = mr $$

In this case, it appears that the conclusion to be proved has been used, and the proof is invalid because it is simply a case of circular reasoning.

Is everybody wrong, or have I missed something?

EDIT: If the proof is in fact broken, can someone please suggest how it could be fixed?

Jackie
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    If you do not have any problem going from the first line to the second line, then you shouldn't have one going from the fifth to the sixth, it is the same argument – caverac Jul 31 '18 at 11:06
  • You should give the domains of $r,x.$ E.g. your 'proof' is wrong for $x=-2, r=2$ – gammatester Jul 31 '18 at 11:08

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I see your point but in the fifth to sixth line the difference is that we know that "$x$" is in fact "$b$" which is the same as the base.

The point is that the definition of logarithm says

$a^x=y\iff x=\log_{a}y$ ($=\log_{a}a^x$)

It is a bit like saying $\log_{10}{1000}=3$ and that you don't need any fancy properties; it follows from the definition.

daruma
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  • This is still conceptually difficult for me. If the property to be proved is also specified by definition, then the proof must be trivial: i.e, $$\log_b(x^r) = r \cdot \log_b(x) & \text{Q.E.D}$$ Surely that can't be?! – Jackie Jul 31 '18 at 12:40
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    No the point is that $\log_b{x^r}$ for general $x$ is non-trivial but if we have $\log_{b}{b^r}$ that is equal to $r$ by the defintion. (The important part is $a^x=y\iff x=\log_a y=\underline{\log_{a}{a^x}})$ – daruma Jul 31 '18 at 13:21