Year11 MATH 3-2-4 Exponential and Logarithmic Functions

Unit2

Focus: Index laws, log laws, and solving equations.

1. Simplify $(2^3)^2 \times 2^{-4}$.
2. Solve for $x$: $3^{x+1} = 27$.
3. Write $\log_2(16) = 4$ in index form.
4. Solve for $x$: $\log_{10}(x) = 2$.
5. Simplify $\log_a(x) + \log_a(y) – \log_a(z)$.
6. Solve $e^x = 5$ (leave in exact form).
7. Simplify $\ln(e^3)$.
8. Solve $2^{2x} – 5(2^x) + 4 = 0$ (Hint: let $u = 2^x$).
9. Differentiate $y = e^{2x}$ (Note: This is usually early Unit 3, but often introduced in Unit 2 “Methods” prep).
10. If $\$1000$ is invested at $5\%$ p.a. compounding annually, write the formula for the amount $A$ after $t$ years.

Answers:

1. 4 ($2^2$) | 2. $x = 2$ | 3. $2^4 = 16$ | 4. $x = 100$ | 5. $\log_a(\frac{xy}{z})$ | 6. $x = \ln(5)$ | 7. 3 | 8. $x = 0, x = 2$ | 9. $2e^{2x}$ | 10. $A = 1000(1.05)^t$.