Let's go through the three questions one by one.

Question 1:

What does ln 1 = (Natural log of 1)?

• The natural logarithm of 1 is 0 because $\ln(1) = 0$. The logarithm function asks, "To what power should we raise $e$ to get 1?" Since $e^0 = 1$, the answer is 0.

Question 2:

Solve for the unknown: $\ln(e^7) = x$

Using logarithmic properties: $\ln(e^7) = 7 \cdot \ln(e) = 7 \cdot 1 = 7$ So, $x = 7$.

Question 3:

Solve for the unknown: $\log_4 8 = \log_4 x$

Since the logs have the same base, we can equate the arguments: $8 = x$ Thus, $x = 8$.

Would you like further details or clarification on any of these solutions? Here are 5 related questions to explore further:

1. What is the natural logarithm of a number close to 0, like $\ln(0.1)$?
2. What happens when you apply the natural logarithm to negative numbers?
3. Can the base of a logarithm in a given problem be a fractional number like 1/2?
4. How do you solve logarithmic equations with different bases?
5. What is the derivative of a logarithmic function like $\ln(x)$?

Tip: Remember, logarithmic functions can simplify exponentials, and they are handy for solving equations involving powers!