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$\left(\log _{10}\left(x\right)\right)^{\log _{10}\left(\log _{10}\left(x\right)\right)} = 10000$

Solve for $x$. What I did first was to set $u = \log_{10} x$ and then try to solve for $u$. However, I got stuck a bit after will simplification and stuff like that. Any help would be appreciated.

Sebastiano
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Eddie Wang
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  • Are we given a value for this expression? Otherwise, what does it mean to "solve for $x$" here? – lulu Feb 18 '23 at 17:59
  • That is not an equation (there is no $=$). Note that $y^{\log_b(y)} = b^{\left(\log_b(y)^2\right)}$ – Henry Feb 18 '23 at 17:59

1 Answers1

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Taking logs of both sides, we have

$$4 = \log_{10}(\log_{10}(x)) \cdot \log_{10}(\log_{10}(x))$$

$$ = \bigl(\log_{10}(\log_{10}(x))\bigr)^2.$$

This means

$$\log_{10}(\log_{10}(x)) = \pm2,$$

and thus,

$$x = 10^{10^{\pm2}}.$$

Luca Armstrong
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