1. ## a log question

How do I answer this log equation?

$log_{2} x + log_{x} 2= 2
$

2. Originally Posted by shawli
How do I answer this log equation?

$log_{2} x + log_{x} 2= 2
$
Use the change of base rule on both parts

$\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$

So that $\log_2(x) = \frac{\ln (x)}{\ln(2)}$

3. Hello, shawli!

Another approach . . .

Thereom: . $\log_ba \:=\:\frac{1}{\log_ab}$

$\log_2 x + \log_x 2\:=\: 2$

We are given: . $\log_2x + \log_x2 \:=\:2$

Use theorem: . $\log_2x + \frac{1}{\log_2x} \:=\:2$

Multiply by $\log_2x\!:\;\;(\log_2x)^2 + 1 \:=\:2\log_2x \quad\Rightarrow\quad (\log_2x)^2 - 2\log_2x + 1 \:=\:0$

Factor: . $\left(\log_2x - 1\right)^2 \:=\:0\quad\Rightarrow\quad \log_2x -1 \:=\:0$

Therefore: . $\log_2x \:=\:1 \quad\Rightarrow\quad x \:=\:2$

4. Thanks!