[Math] Log power rule problem

exponentiationlogarithmsplane-curves

According to many parts of the Internet, this log rule is used.

log(a^b) = b*log(a)

The proof is:

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Now let's say I want to use the rule in a Cartesian plane. I currently have the function…

f(x) = 2*log(x)

I find the domain and it turns out to be…

x > 0

Now since I can use the Log Power Rule for rearranging the question, nothing should change, right?

f(x) = log(x^2)

Checks the domain…

ALL REAL X

This is not right…

Did I do something wrong in the process, or is the Log Power Rule not very correct?

Edit: The answer stated that it is not meant to be used in this way. So is there a version of the Log Power Law that works correctly with negative numbers?

Best Answer

First, the domain of the first expression is $x > 0$; when $x = 0$, log is undefined.

Second, the log power rule proof starts out with "write $x = a^m$", but this is generally only possible if $x$ is positive. (It's possible for negative $x$, but only when $a$ is negative, and that's not one of the assumptions of the proof.) So the log power rule should be taken as true only in the cases for which you've proved it. For instance, it's not true that $$ \log 4 = \log ( (-2)^2 ) = 2 \log (-2) $$ because the thing on the right is undefined.

Short summary:

  • The log power rule is fine, but holds true only when the expressions on both sides are defined. The place you learned it was apparently not cautious enough to state this, or you failed to notice it.
  • The software you're using is incorrectly suggesting that $x = 0$ is in the domain of the function $x \mapsto \log x$.
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