Logarithmic equation $\log^2_{4}{x}-\log_{2}{x}+4=0$

Question

How to solve this logarithmic equation?

$\log^2_{4}{x}-\log_{2}{x}+4=0$

I do not understand how to start a solution.

Answer 1

$$\log^2_{4}{x}-\log_{2}{x}+4=0$$

$$\log^2_{4}{x}-2\log_{4}{x}+4=0$$

Let $$u=\log_{4}{x}$$

We get $$u^2-2u+4=0$$

Which does not have real solutions.

Answer 2

• HINT: $ \log_{a^{c}}{b} = \frac{1}{c}\log_{a}{b} \ $ and $\log_{2}{x} = t$

Answer 3

The expression is equivalent to $\frac{log^2(x)}{log^2(4)}-\frac{log^2(x)}{log^2(2)}+4=0$ but like you've been told, does not have real solutions. Actually the complex solutions are $x=2^{2-2i\sqrt{3}}$ and $x=2^{2+2i\sqrt{3}}$