3 galore!

Algebra Level 3

x log ⁡ 3 x = 81 x 3 \displaystyle \large x^{\log_3 x} = 81x^3

There are two solutions x 1 x_1 and x 2 x_2 to the above equation, with x 1 > x 2 x_1 > x_2 . Find x 1 x 2 \dfrac{x_1}{x_2} .


The answer is 243.

Zach Abueg by Zach Abueg , 22, USA

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Chew-Seong Cheong
Jul 27, 2017

x log ⁡ 3 x = 81 x 3 log ⁡ 3 x ⋅ log ⁡ 3 x = log ⁡ 3 81 + 3 log ⁡ 3 x log ⁡ 3 2 x = 4 + 3 log ⁡ 3 x log ⁡ 3 2 x − 3 log ⁡ 3 x − 4 = 0 ( log ⁡ 3 x − 4 ) ( log ⁡ 3 x + 1 ) = 0 ⟹ x = { log ⁡ 3 x = 4 ⟹ x 1 = 3 4 = 81 log ⁡ 3 x = − 1 ⟹ x 2 = 3 − 1 = 1 3 ⟹ x 1 x 2 = 81 1 3 = 243 \begin{aligned} x^{\log_3 x} & = 81x^3 \\ \log_3 x \cdot \log_3 x & = \log_3 81 + 3 \log_3 x \\ \log_3^2 x & = 4 + 3 \log_3 x \\ \log_3^2 x - 3 \log_3 x - 4 & = 0 \\ (\log_3 x -4)(\log_3 x + 1) & = 0 \\ \implies x & = \begin{cases} \log_3 x = 4 & \implies x_1 = 3^4 = 81 \\ \log_3 x = -1 & \implies x_2 = 3^{-1} = \frac 13 \end{cases} \\ \implies \frac {x_1}{x_2} & = \frac {81}{\frac 13} = \boxed{243} \end{aligned}

8 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...