Help me to solve this. Efforts written in photo.

Question

If $$ 2^{\log5} × 5^{\log2}= 2^{\log x} $$ Find $$ \log_5 {x^{2/3}} $$

Here my efforts in the picture:

Answer 1

Hint: $a^{\log b}=b^{\log a}$.

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From my hint, you have

$$2^{\log5}\cdot5^{\log2}=2^{\log x}$$

$$2^{\log5}\cdot2^{\log5}=2^{\log x}$$

This gives

$$2\log5=\log x$$

Thus

$$x=25$$

Substitution gives

$$\log_525^{\frac23}=\frac43$$

Answer 2

$$\log _5(x^{\frac{2}{3}} )=\frac{2}{3}\log _5 (x)$$ because assuming $x>0$, $\log _a\left(x^b\right)=b\cdot \log _a\left(x\right)$.

$$2^{\log _{10}\left(5\right)}\cdot \:5^{\log _{10}\left(2\right)}=2^{\log _{10}\left(x\right)}$$

$$\ln \left(2^{\log _{10}\left(5\right)}\cdot \:5^{\log _{10}\left(2\right)}\right)=\ln \left(2^{\log _{10}\left(x\right)}\right)$$ $$\ln \left(2^{\log _{10}\left(5\right)}\cdot \:5^{\log _{10}\left(2\right)}\right)=\log _{10}\left(x\right)\ln \left(2\right) \rightarrow \log _{10}\left(x\right)\ln \left(2\right)=\ln \left(2^{\log _{10}\left(5\right)}\cdot \:5^{\log _{10}\left(2\right)}\right)$$

Hence

$$\log _{10}\left(x\right)=\frac{\ln \left(2^{\log _{10}\left(5\right)}\cdot \:5^{\log _{10}\left(2\right)}\right)}{\ln \left(2\right)}=\log _{10}\left(10^{\frac{\ln \left(2^{\log _{10}\left(5\right)}\cdot \:5^{\log _{10}\left(2\right)}\right)}{\ln \left(2\right)}}\right) \Longrightarrow $$$$x=10^{\dfrac{\ln \left(2^{\log _{10}\left(5\right)}\cdot 5^{\log _{10}\left(2\right)}\right)}{\ln \left(2\right)}}$$