Without calculator...#1

Algebra Level 2

Solve for $x:$

$\large \log_{x} {32} = 64$

Give your answer to the nearest whole number.

If you are writing a solution, give $x$ in exact form.

1 3 2 0

2 solutions

Jordan Cahn
Nov 28, 2018

$\begin{aligned} \log_x 32 &= 64 \\ x^{\log_x 32} &= x^{64} \\ 32 &= x^{64} \\ x &= \sqrt[64]{32} \end{aligned}$

Note that $\sqrt[5]{32} = 2$ , so $1<\sqrt[64]{32}< \sqrt[10]{32} = \sqrt{2} < 1.5$ . Thus the closest integer to $x$ is $\boxed{1}$ .

Mohammad Farhat
Jun 11, 2019

With Wolfram Alpha I get $2^{\frac{5}{64}}$