Solve logx^-1+log^-1x/logde=-3/2

$$\frac { \log x ^ { - 1 } + \log ^ { - 1 } x } { \log d e } = - \frac { 3 } { 2 }$$

Solve for d

$$d=e^{-\frac{2\ln(10)^{2}}{3\ln(x)}}x^{-\frac{1}{\ln(x)}+\frac{2}{3}}$$

$$x>0\text{ and }x\neq 1\text{ and }x\neq \frac{1}{10}\text{ and }x\neq 10\text{ and }\left(e^{-\frac{2\ln(10)^{2}}{3\ln(x)}}x^{-\frac{1}{\ln(x)}+\frac{2}{3}}-\frac{1}{e}>0\text{ or }e^{-\frac{2\ln(10)^{2}}{3\ln(x)}}x^{-\frac{1}{\ln(x)}+\frac{2}{3}}>0\right)\text{ and }e^{-\frac{2\ln(10)^{2}}{3\ln(x)}}x^{-\frac{1}{\ln(x)}+\frac{2}{3}}-\frac{1}{e}\neq 0\text{ and }e^{-\frac{2\ln(10)^{2}}{3\ln(x)}}x^{-\frac{1}{\ln(x)}+\frac{2}{3}}\neq \frac{1}{e}$$

Solve for x

$$\left\{\begin{matrix}x=d^{\frac{3}{4}}e^{\frac{\sqrt{9\ln(d)^{2}+18\ln(d)+16\ln(10)^{2}+9}+3}{4}}\text{, }&d^{\frac{3}{4}}e^{\frac{\sqrt{9\ln(d)^{2}+18\ln(d)+16\ln(10)^{2}+9}+3}{4}}\neq 1\text{ and }d>0\text{ and }d\neq \frac{1}{e}\\x=d^{\frac{3}{4}}e^{\frac{-\sqrt{9\ln(d)^{2}+18\ln(d)+16\ln(10)^{2}+9}+3}{4}}\text{, }&d^{\frac{3}{4}}e^{\frac{-\sqrt{9\ln(d)^{2}+18\ln(d)+16\ln(10)^{2}+9}+3}{4}}\neq 1\text{ and }d>0\text{ and }d\neq \frac{1}{e}\end{matrix}\right.$$