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Prove $x^{\ln(x)} = e^{(\ln(x))^3}$

($x$ raised to $\ln(x)$ $=$ $e$ raised to ($\ln(x)$ cubed).

I have an exam in 20 minutes. Appreciate the answers.

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closed as off-topic by user127.0.0.1, user91500, Davide Giraudo, Avitus, Umberto P. Apr 7 at 13:22

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4 Answers 4

up vote 1 down vote accepted

As usual, it may be the question is poorly posted. It may have been "solve the equation $\;x^{\log x}=e^{\log^3x}\;$" , and then

$$x^{\log x}=e^{\log^2x}\implies x^{\log x}=e^{\log^3x}\iff \log^2x=\log^3x\iff$$

$$\log^2x(\log x-1)=0\iff\begin{cases}\log x=0\iff x=1\\\text{or}\\\log x=1\iff x=e\end{cases}$$

Oh, well: it is now more than 20 minutes after the OP asked and the exam is over almost surely...

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Perhaps I'm mistaken but I believe it should be squared :$x^{\ln(x)}=(e^{\ln(x)})^{\ln(x)}=e^{(\ln(x))^{2}}$.

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No, it isn't that. The question is correct –  TheEconomist Jan 19 at 5:12
@TheEconomist Apparently not. –  T. Bongers Jan 19 at 5:14
If the question's correct then it is a rather pretty easy one. It's answer is: the equality is wrong. –  DonAntonio Jan 19 at 9:51
Why does this question deserve 5 downvotes? The question was correct. –  TheEconomist Jan 21 at 12:20
@TheEconomist, if the OP was correct, the only possible answer is: the equality is wrong...but you did not asked whether the equality is right or wrong, but to prove that equality, which of course cannot be done. –  DonAntonio Jan 21 at 15:46

There is nothing to prove the two expressions are not equivilent, this can be seen by taking the logarithm of both sides.

$$\ln(x)+\ln(\ln(x))\ne \ln(x)^3$$

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Taking the logarithm on both side you should find that $\text{ln}(x)^2 = \text{ln}(x)^3$ , which has two solutions. –  Olivier Jan 19 at 5:44
@Olivier User71352 changed the question, look at the un-edited version –  Ethan Jan 19 at 5:46

It is squared. Here is a step by step explanation of user71352's answer.

For any $a$

$$ x^a = e^{a \ln(x)}$$ Put $a = \ln(x)$ and you get $$ x^{\ln(x)} = e^{\ln(x) \cdot \ln(x)} = e^{\ln(x)^2}$$

Verification using my calculator:

$$ 3^{\ln(3)} \approx 3.3433 \\ e^{\ln(3)^2} \approx 3.3433 $$

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