To convert the logarithm $\log_7(15)$ to base $e$, we use the change of base formula for logarithms:

$\log_b(a) = \frac{\ln(a)}{\ln(b)}$

Here, $b = 7$ and $a = 15$. So:

$\log_7(15) = \frac{\ln(15)}{\ln(7)}$

Now, using approximate values for the natural logarithms:

$\ln(15) \approx 2.7081 \quad \text{and} \quad \ln(7) \approx 1.9459$

$\log_7(15) \approx \frac{2.7081}{1.9459} \approx 1.391$

Thus, $\log_7(15)$ to base $e$ is approximately 1.391.

Would you like further details or have any questions?

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Here are some related questions you might find interesting:

1. What is the change of base formula for logarithms?
2. How do you derive the natural logarithm ($\ln$) of a number?
3. Can you calculate $\log_7(20)$ to base $e$?
4. What is the significance of natural logarithms in mathematics?
5. How can logarithmic identities simplify complex logarithmic expressions?

Tip: The change of base formula is very useful when you need to compute logarithms with a calculator that only supports base 10 or base $e$.