give me 10 questions of solving for x with natural logarithms with base e

Here are 10 questions involving solving for \( x \) using natural logarithms with base \( e \) (denoted by \( \ln \)):

1. \( \ln(x) = 3 \)
2. \( \ln(2x) = 5 \)
3. \( \ln(x) + \ln(3) = 4 \)
4. \( 2\ln(x) = 6 \)
5. \( \ln(x - 1) = 2 \)
6. \( \ln(x^2) = 8 \)
7. \( \ln(e^x) = 7 \)
8. \( \ln(x + 3) = 2\ln(2) \)
9. \( \ln(4x) - \ln(2) = 1 \)
10. \( e^{\ln(x)} = 10 \)

Would you like details on solving any of these, or do you have any questions?

Here are five related questions to expand the topic:

1. Solve for \( x \) in \( \ln(5x) = 7 \).
2. Solve for \( x \) in \( \ln(x - 2) = 3 \).
3. Solve for \( x \) in \( 3\ln(x) = 9 \).
4. Solve for \( x \) in \( \ln(2x + 1) = 4 \).
5. Solve for \( x \) in \( \ln(x^3) = 9 \).

**Tip:** Always remember the inverse relationship between \( e^x \) and \( \ln(x) \), which helps in simplifying these equations.

Practice solving for x using natural logarithms with base e (ln) through 10 detailed questions. This exercise covers essential logarithmic properties and their inverse relationship with exponents. Suitable for students in Grades 10-12.

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