Solve 2^m=3quad3^n=2quadb^m=?

$$2 ^ { m } = 3 \quad 3 ^ { n } = 2 \quad b ^ { m } = ?$$

Evaluate (complex solution)

$$2^{m}=3^{n+1}\text{ and }3^{n+1}=2b^{m}$$

Solve for n

$$m=\log_{2}\left(3\right)n+\log_{2}\left(3\right)\text{ and }\left(\frac{e^{\frac{\ln(2)^{2}}{\ln(3)}}}{2\times 2^{n}}\right)^{-\frac{1}{n+1}}-2\neq 0\text{ and }n\neq -1\text{ and }\left(\frac{e^{\frac{\ln(2)^{2}}{\ln(3)}}}{2\times 2^{n}}\right)^{-\frac{1}{n+1}}\neq 2\text{ and }b=\left(\frac{e^{\frac{\ln(2)^{2}}{\ln(3)}}}{2\times 2^{n}}\right)^{-\frac{1}{n+1}}$$

Solve for m

$$\left\{\begin{matrix}m=\log_{2}\left(3\right)\left(n+1\right)\text{, }&b=\left(e^{-\frac{\ln(2)^{2}}{\ln(3)}}\times 2^{n+1}\right)^{\frac{1}{n+1}}\text{ and }n\neq -1\text{ and }n\neq \log_{3}\left(2\right)-1\\m=1\text{, }&n=\log_{3}\left(2\right)-1\text{ and }b=1\end{matrix}\right.$$