We are given the equation $2e^{3x} = 5e^{8x}$ and asked to solve for $x$.

Step-by-step solution:

1. Rewrite the equation: $2e^{3x} = 5e^{8x}$

2. Divide both sides by $e^{3x}$: $\frac{2e^{3x}}{e^{3x}} = \frac{5e^{8x}}{e^{3x}}$ Simplifying: $2 = 5e^{5x}$ (since $e^{8x} / e^{3x} = e^{(8x - 3x)} = e^{5x}$)

3. Solve for $e^{5x}$: $e^{5x} = \frac{2}{5}$

4. Take the natural logarithm of both sides: $\ln(e^{5x}) = \ln\left(\frac{2}{5}\right)$

5. Simplify using the logarithmic property $\ln(e^y) = y$: $5x = \ln\left(\frac{2}{5}\right)$

6. Solve for $x$: $x = \frac{1}{5} \ln\left(\frac{2}{5}\right)$

Thus, the exact solution is: $x = \frac{1}{5} \ln\left(\frac{2}{5}\right)$

Would you like further details or have any questions? Here are some related questions you might find helpful:

1. How do logarithmic properties simplify exponential equations?
2. What are some other methods to solve exponential equations without a calculator?
3. How does the natural logarithm function relate to the exponential function?
4. Can this equation have multiple solutions? Why or why not?
5. What are common mistakes when solving exponential equations?

Tip: Always use logarithmic identities to simplify exponential expressions before applying numerical methods.