11.6 Exponential and Logarithmic Functions

**Index laws.** Simplify each expression as a single power of $x$. (a) $x^4 \cdot x^3$ (b) $\dfrac{x^9}{x^4}$ (c) $(x^2)^5$ (d) $\left(\dfrac{1}{x^3}\right)^{-2}$

**Fractional indices [EXT].** Evaluate exactly. (a) $25^{1/2}$ (b) $8^{2/3}$ (c) $16^{3/4}$ (d) $\left(\dfrac{1}{27}\right)^{-2/3}$

**Compound interest.** CHF 5000 is invested at 4.5% per year compounded annually. Let $V$ (CHF) be the value after $t$ years. (a) Write a formula for $V$ in terms of $t$. (b) Find the value after 8 years to the nearest franc. (c) Find the smallest integer $t$ for which the investment has at least doubled. (d) By what percentage has the investment grown after 25 years?

**Café customer model.** A café records weekly customers $C$ (in hundreds). At $t = 1$ week, $C = 6$; at $t = 4$ weeks, $C = 162$. Model $C(t) = a \cdot b^t$ with $a, b > 0$. (a) Set up two equations and find $a$ and $b$. (b) State $C(0)$ in customers. (c) Sketch $C(t)$ for $0 \leq t \leq 4$, marking the $C$-intercept and $C(4)$. (d) Find, using logarithms, the time at which $C$ first reaches 1000 (i.e. 10 hundred).

**Exponential graph features.** Let $f(x) = 2^x$. (a) State the domain, range, $y$-intercept, and horizontal asymptote of $f$. (b) On the same axes, sketch $y = 2^x$ and $y = 2^{x - 3} + 1$. (c) State the $y$-intercept and asymptote of $y = 2^{x - 3} + 1$.

**Half-life [EXT].** A radioactive isotope has half-life 8 years. (a) Write a model $M(t) = M_0 \cdot k^t$ for the mass after $t$ years; state $k$ exactly. (b) After 24 years, what fraction of the original mass remains? (c) Find $t$ (to 3 s.f.) for 10% of the original mass to remain.

**Logarithm equation — extraneous root [EXT].** Solve $\log_3 x + \log_3(x - 2) = 1$ and explain why one algebraic candidate must be rejected.

**Hidden quadratic [EXT].** Solve $4^x - 5 \cdot 2^x + 4 = 0$ for real $x$.

**Two-point exponential model.** A population satisfies $P(t) = a \cdot b^t$ with $P(2) = 18$ and $P(5) = 486$. Find $a$ and $b$, and predict $P(8)$.

**Depreciation.** A new car costs CHF 32 000 and depreciates at 18% per year. (a) Write a model for the value $V(t)$ after $t$ years. (b) Find the value after 5 years, to the nearest franc. (c) Find the year in which the car is first worth less than CHF 10 000 (use logs).

**Log laws [EXT].** Express each as a single logarithm (assume all arguments positive). (a) $\log a + \log b - \log c$ (b) $2 \log p - 3 \log q$ (c) $\dfrac{1}{2} \log m + \log n$

**Modelling — compound interest with monthly compounding.** CHF 1000 is invested at a nominal 6% annual rate. (a) Find the value after 1 year if interest is compounded **annually**. (b) Find the value after 1 year if interest is compounded **monthly**. (c) Find the effective annual rate (EAR) for monthly compounding, to 3 s.f. (d) For how many years would CHF 1000 take to **double** under monthly compounding (to 3 s.f.)?