Question Number 175685 by Linton last updated on 05/Sep/22
$$\mathrm{3}^{\mathrm{5}^{{x}} } =\:\mathrm{5}^{\mathrm{3}^{{x}} } \\ $$$${solve}\:{for}\:{x} \\ $$
Answered by mahdipoor last updated on 05/Sep/22
$${ln}\left({ln}\left(\mathrm{3}^{\mathrm{5}^{{x}} } \right)\right)={ln}\left({ln}\left(\mathrm{5}^{\mathrm{3}^{{x}} } \right)\right)\:\Rightarrow \\ $$$${ln}\left(\mathrm{5}^{{x}} ×{ln}\left(\mathrm{3}\right)\right)={ln}\left(\mathrm{3}^{{x}} ×{ln}\left(\mathrm{5}\right)\right)\:\Rightarrow \\ $$$${x}.{ln}\mathrm{5}+{ln}\left({ln}\mathrm{3}\right)={x}.{ln}\mathrm{3}+{ln}\left({ln}\mathrm{5}\right)\:\Rightarrow \\ $$$${x}=\frac{{ln}\left({ln}\mathrm{5}\right)−{ln}\left({ln}\mathrm{3}\right)}{{ln}\mathrm{5}−{ln}\mathrm{3}}=\frac{{ln}\left(\frac{{ln}\mathrm{5}}{{ln}\mathrm{3}}\right)}{{ln}\left(\frac{\mathrm{5}}{\mathrm{3}}\right)}= \\ $$$${log}_{\frac{\mathrm{5}}{\mathrm{3}}} \left({log}_{\mathrm{3}} \mathrm{5}\right) \\ $$
Answered by Rasheed.Sindhi last updated on 06/Sep/22
$$\mathrm{3}^{\mathrm{5}^{{x}} } =\:\mathrm{5}^{\mathrm{3}^{{x}} } \\ $$$$\mathrm{5}^{{x}} \mathrm{log}\:\mathrm{3}=\mathrm{3}^{{x}} \mathrm{log}\:\mathrm{5} \\ $$$$\left(\frac{\mathrm{5}}{\mathrm{3}}\right)^{{x}} =\frac{\mathrm{log}\:\mathrm{5}}{\mathrm{log}\:\mathrm{3}}=\mathrm{log}_{\mathrm{3}} \mathrm{5} \\ $$$${x}\left(\mathrm{log5}−\mathrm{log3}\right)=\mathrm{log}\left(\mathrm{log}_{\mathrm{3}} \mathrm{5}\right) \\ $$$${x}=\frac{\mathrm{log}\left(\mathrm{log}_{\mathrm{3}} \mathrm{5}\right)}{\mathrm{log5}−\mathrm{log3}}\:\:\: \\ $$
Commented by Rasheed.Sindhi last updated on 05/Sep/22
$$\mathcal{T}{hank}\:{you}\:{for}\:{correction}\:{sir}! \\ $$