The natural logarithm uses base e and e^ln x simplifies to x, so this Year 12 sheet opens with the two inverse facts that make logarithms work. The derivative answers list eˣ and 1/x, and ln a + ln b is given as the product rule.
Key Takeaways
The natural logarithm has base e (Q1).
e^ln x = x (Q2).
Q3 records ln a + ln b for logarithms of products.
Section A: Multiple Choice Questions (1 Mark Each)
Choose the correct option for each question.
1.The natural logarithm is the logarithm to the base
(a) e(b) 10(c) 2(d) π
2.The value of e^ln x is
(a) x(b) 1(c) e(d) ln x
3.ln(ab) is equal to
(a) ln a + ln b(b) ln a − ln b(c) ln a × ln b(d) a ln b
4.The derivative of eˣ with respect to x is
(a) eˣ(b) x eˣ(c) 1/x(d) ln x
5.The derivative of ln x with respect to x is
(a) 1/x(b) x(c) ln x(d) eˣ
6.The value of ln 1 is
(a) 0(b) 1(c) e(d) −1
Section B: Short Answer Type Questions (2 Marks Each)
Show all steps clearly.
7.State the value of e correct to 2 decimal places.
8.Write the logarithm rules for ln(ab), ln(a/b) and ln(aⁿ).
9.What is the derivative of eˣ?
10.What is the derivative of ln x?
11.Simplify e^ln 5.
12.Solve ln x = 0.
13.Write e³ ≈ 20.09 in logarithmic form.
14.How do you model exponential growth mathematically?
Section C: Numericals & Word Problems (3 Marks Each)
Apply the concepts to solve the problems. Show all working.
15.Solve eˣ = 7. Using x = ln 7, give your answer correct to 2 decimal places.
16.Solve ln x = 2. Using x = e², give your answer correct to 2 decimal places.
17.Simplify ln 8 − ln 2 using the logarithm rule ln(a/b).
18.Simplify 2 ln 3 using the power rule of logarithms.
19.Solve e²ˣ = 10 for x, giving your answer correct to 2 decimal places.
20.A population grows by 5% per year. Using the model P = P₀(1.05)ᵗ, state the growth multiplier.
Answer Key
1. a) e
2. a) x
3. a) ln a + ln b
4. a) eˣ
5. a) 1/x
6. a) 0
7. Refer to solution guide.
8. Refer to solution guide.
9. Refer to solution guide.
10. Refer to solution guide.
11. Refer to solution guide.
12. Refer to solution guide.
13. Refer to solution guide.
14. Refer to solution guide.
15. Refer to solution guide.
16. Refer to solution guide.
17. Refer to solution guide.
18. Refer to solution guide.
19. Refer to solution guide.
20. Refer to solution guide.
End of Worksheet
People Also Ask
What is the base of the natural logarithm?
e, option (a) in this worksheet (Q1).
What is the value of e^ln x?
x, option (a) in this worksheet (Q2).
What is ln a + ln b equal to?
ln(ab) — recorded as ln a + ln b, option (a) in this worksheet (Q3).
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