To solve the equation $e^{5x} = 10$, follow these steps:

Part 1: Exact Solution

1. Start by taking the natural logarithm of both sides to get rid of the exponent: $\ln(e^{5x}) = \ln(10)$
2. Using the property $\ln(e^a) = a$, simplify the left side: $5x = \ln(10)$
3. Divide both sides by 5 to solve for $x$: $x = \frac{\ln(10)}{5}$

So, the exact solution is: $x = \frac{\ln(10)}{5}$

Part 2: Approximate Solution

1. Calculate $\ln(10) \approx 2.3026$.
2. Divide by 5: $x \approx \frac{2.3026}{5} \approx 0.4605$

So, the approximate solution to four decimal places is: $x \approx 0.4605$

Let me know if you'd like further clarification on any steps!

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Further Questions:

1. How does the natural logarithm function help in solving exponential equations?
2. What would the solution be if the equation were $e^{5x} = 20$?
3. How can this method be applied to equations with other bases, like $2^{5x} = 10$?
4. Why is it important to keep as many decimal places as possible until the final rounding?
5. How would you verify this solution?

Tip: Remember, the natural logarithm $\ln(x)$ is the inverse of $e^x$, making it a powerful tool to solve equations with $e$ as the base.