Question

$$1\ \log_{2}x+\log_{x}2=2\ f\cdot dx$$

Solve for d (complex solution)

$\left\{\begin{matrix}d=\frac{\ln(x)^{2}+\ln(2)^{2}}{2\ln(2)fx\ln(x)}\text{, }&f\neq 0\text{ and }x\neq 1\text{ and }x\neq 0\\d\in \mathrm{C}\text{, }&f=0\text{ and }x=2^{i}\end{matrix}\right.$

Solve for f (complex solution)

$\left\{\begin{matrix}f=\frac{\ln(x)^{2}+\ln(2)^{2}}{2\ln(2)dx\ln(x)}\text{, }&d\neq 0\text{ and }x\neq 1\text{ and }x\neq 0\\f\in \mathrm{C}\text{, }&d=0\text{ and }x=2^{i}\end{matrix}\right.$

Solve for d

$d=\frac{\ln(x)^{2}+\ln(2)^{2}}{2\ln(2)fx\ln(x)}$
$f\neq 0\text{ and }x\neq 1\text{ and }x>0$

Solve for f

$f=\frac{\ln(x)^{2}+\ln(2)^{2}}{2\ln(2)dx\ln(x)}$
$d\neq 0\text{ and }x\neq 1\text{ and }x>0$