1. **State the problem:** We need to find the value of $x$ given the equation $\log_e 2 \cdot \log_x 625 = \log_{10} 16 \cdot \log_2 10$. 2. **Rewrite the logarithms in a common base:** Recall that $\log_a b = \frac{\log_c b}{\log_c a}$ for any positive base $c \neq 1$. 3. Express each logarithm in terms of natural logarithm $\ln$ (log base $e$): $$\log_e 2 = \ln 2$$ $$\log_x 625 = \frac{\ln 625}{\ln x}$$ $$\log_{10} 16 = \frac{\ln 16}{\ln 10}$$ $$\log_2 10 = \frac{\ln 10}{\ln 2}$$ 4. Substitute these into the equation: $$\ln 2 \cdot \frac{\ln 625}{\ln x} = \frac{\ln 16}{\ln 10} \cdot \frac{\ln 10}{\ln 2}$$ 5. Simplify the right side by canceling $\ln 10$: $$\frac{\ln 16}{\cancel{\ln 10}} \cdot \frac{\cancel{\ln 10}}{\ln 2} = \frac{\ln 16}{\ln 2}$$ 6. So the equation becomes: $$\ln 2 \cdot \frac{\ln 625}{\ln x} = \frac{\ln 16}{\ln 2}$$ 7. Multiply both sides by $\ln x$: $$\ln 2 \cdot \ln 625 = \frac{\ln 16}{\ln 2} \cdot \ln x$$ 8. Divide both sides by $\frac{\ln 16}{\ln 2}$: $$\ln x = \frac{\ln 2 \cdot \ln 625}{\frac{\ln 16}{\ln 2}} = \ln 2 \cdot \ln 625 \cdot \frac{\ln 2}{\ln 16}$$ 9. Simplify the right side: $$\ln x = \frac{(\ln 2)^2 \cdot \ln 625}{\ln 16}$$ 10. Express $625$ and $16$ as powers of 5 and 2 respectively: $$625 = 5^4 \Rightarrow \ln 625 = 4 \ln 5$$ $$16 = 2^4 \Rightarrow \ln 16 = 4 \ln 2$$ 11. Substitute these back: $$\ln x = \frac{(\ln 2)^2 \cdot 4 \ln 5}{4 \ln 2}$$ 12. Cancel 4 and one $\ln 2$: $$\ln x = \cancel{4} \cdot (\ln 2)^2 \cdot \ln 5 / (\cancel{4} \cdot \ln 2) = \ln 2 \cdot \ln 5$$ 13. So: $$\ln x = \ln 2 \cdot \ln 5$$ 14. To find $x$, exponentiate both sides with base $e$: $$x = e^{\ln 2 \cdot \ln 5}$$ **Final answer:** $$\boxed{x = e^{\ln 2 \cdot \ln 5}}$$