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Consider the following,

$$\ln\left|\frac{(x-2)^{\frac{2}{3}}}{(x+3)^{\frac{1}{3}}}\right|\neq \ln\left|\frac{(x-2)^{\frac{1}{3}}}{(x+3)}\right|$$

The simplified answer here maintains the exponents in the numerator and denominator of the fraction inside the natural log.

In a regular fraction we would be able to subtract the exponent of the numerator by the exponent in the denominator, but in this instance there is something about the nature of the natural log that inhibits this algebraic process. What feature of the natural log prevents us from doing this? Is there a formal proof for why this does not work?

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    $\begingroup$ It is not because of the logarithm, you cannot substract exponents because $x-2 \neq x+3$. $\endgroup$ Commented Jan 26, 2025 at 22:24
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    $\begingroup$ You can only subtract the exponents if the bases are the same - careful! $\endgroup$ Commented Jan 26, 2025 at 22:32
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    $\begingroup$ Even if the bases were the same you couldn’t “cancel” like this. You increased the exponent of the denominator from $1/3$ to $1.$ $\endgroup$ Commented Jan 26, 2025 at 22:34

2 Answers 2

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First, $(x-2)^{2/3}$ is not clearly defined if $x<2$. Your absolute values should be around $x-2$ and $x+3$ instead of around the fractions.

That said, your problem has nothing to do with logs: $\frac{|x-2|^{2/3}}{|x+3|^{1/3}}\ne\frac{|x-2|^{1/3}}{|x+3|}$ simply because $$\frac{|x-2|^{2/3}}{|x+3|^{1/3}}\frac{|x+3|}{|x-2|^{1/3}}=|x-2|^{1/3}|x+3|^{2/3}\ne1.$$

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I think it is important to understand why the rules work the way they do: if you have some positive number $b$, and positive integers $m$ and $n$ with $m>n$, then $$\frac{b^m}{b^n}=\frac{\underbrace{b\times b\times \dots\times b}_{m\text{ times}}}{\underbrace{b\times b\times \dots\times b}_{n\text{ times}}}=\underbrace{b\times b\times \dots\times b}_{m-n\text{ times}}=b^{m-n}$$ While if you have something like $$\frac{b^m}{a^n}=\frac{\underbrace{b\times b\times \dots\times b}_{m\text{ times}}}{\underbrace{a\times a\times \dots\times a}_{n\text{ times}}}$$ which one can't simplify really.

Hope this helps. :)

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