solve $\log_3^2(x)-\log_2(x)=2$

Question

The solution for the equation $\log_3^2(x)-\log_3(x)=2$ is

a) $s=\{2,-1\}$; b) $s=\{6,-3\}$; c) $s=\{9, 1/3 \}$; d) $s=\{27, 1/9 \}$; e) $s=\{ 1/6 ,12\}$

It was given in a test at school and I could not solve. I put on wolfram and the answer is not on the options. Link

Yeah, I'm really sorry to inform you that WolframAlpha got it right, and you might have made a typo... — konewka, Oct 16 '14 at 15:00

score 5 · Answer 1 · answered Oct 16 '14 at 15:41

5

From the given options, I guess that you want to solve the following for $x$ : $$(\log_3(x))^2-\log_\color{red}{3}(x)=2.$$ Let $t=\log_3(x)$. Then, we have $$t^2-t-2=0\iff (t-2)(t+1)=0\iff t=2,-1\iff x=3^2,3^{-1}.$$

answered Oct 16 '14 at 15:41

mathlove

139,939

To point out the blatantly obvious: this answer assumes that the OP made a typo, and wrote $\log_2$ instead of $\log_3$. – apnorton Oct 16 '14 at 15:44
@anorton: Thanks:) – mathlove Oct 16 '14 at 15:46

score 2 · Answer 2 · answered Oct 16 '14 at 15:04

2

$\left(\frac{\log(x)}{\log(3)}\right)^2-\left(\frac{\log(x)}{\log(2)}\right)=2$
setting $t=\log(x)$ we get a quadratic equation.

answered Oct 16 '14 at 15:04

Dr. Sonnhard Graubner

95,283

solve $\log_3^2(x)-\log_2(x)=2$

2 Answers2