For the Love of Logs

Algebra Level 3

Determine all real values of $x$ for which

$\large ( \log_{10} x)^{\log_{10}(\log_{10} x)}=10,000$

Express your answers in the form $x=10^{y}$

Select the correct pair of values for $y$

1000, 0.001 10000, 0.0001 100, 0.01 10, 0.1

2 solutions

Chew-Seong Cheong
Mar 14, 2016

If $x = 10^y$ , then $\log_{10} x = y$ , therefore,

$\begin{aligned} (\log_{10} x)^{\log_{10}(\log_{10} x)} & = 10000 \\ y^{\log_{10} y} & = 10^4 \quad \quad \small \color{#3D99F6}{\text{Taking } \log_{10} \text{ both sides,}} \\ \log_{10} y \log_{10} y & = 4 \\ \log_{10}^2 y & = 4 \\ \Rightarrow \log_{10} y & = \pm 2 \\ y & = \begin{cases} 10^2 \\ 10^{-2} \end{cases} = \boxed{100, 0.01} \end{aligned}$

Aditya Narayan Sharma
Mar 14, 2016

Same soln ! Options reduce the difficulty of the sum however to some extent as trial and error gets his role.

For the Love of Logs

2 solutions

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