Answer Key
11.6 Exponential and Logarithmic Functions
Pack A — Answers
| # | Question | Answer |
|---|---|---|
| 1 | Simplify $x^{ 5} \cdot x^{ 3}$ as a single power of $x$. | $x^{8}$ |
| 2 | Simplify $\dfrac{x^{ 8}}{x^{ 3}}$ as a single power of $x$. | $x^{5}$ |
| 3 | Simplify $\left(z^{ 4}\right)^{ 3}$ as a single power. | $z^{12}$ |
| 4 | Write $\left(\dfrac{1}{2}\right)^{-3}$ as a whole number. | 8 |
| 5 | Evaluate $(17)^0$. | 1 |
| 6 | State the $y$-intercept and the equation of the horizontal asymptote of $y = 2^x$. | $y$-intercept $(0, 1)$; asymptote $y = 0$ |
| 7 | Evaluate $ 2^{ 6}$. | 64 |
| 8 | Evaluate $ 49^{1/2}$ exactly. | 7 |
| 9 | Solve $ 2^x = 64$, giving an exact answer. | $x = 6$ |
| 10 | CHF\,1000 is invested at 5% per year, compounded annually. Find the value after 1 year. | CHF 1050 |
| 11 | Write $ 2^{ 5} = 32$ in logarithmic form. | $\log_2 32 = 5$ |
| 12 | Evaluate $\log_{ 5} 125$. | 3 |
| 13 | A radioactive substance decays at 12% per year. Find the multiplier per year, and the amount after 5 years if initial mass is 100 g. | Multiplier 0.88; after 5 years $\approx 52.8$ g |
| 14 | Evaluate $ 8^{2/3}$ exactly. | 4 |
| 15 | A colony of 200 bacteria doubles every hour. Find the population after 4 hours. | 3200 |
| 16 | Simplify $\log 8 + \log 5 - \log 4$. | 1 |
| 17 | A function $y = a \cdot b^x$ has $y$-intercept 5 and passes through $(1, 15)$. Find $a$ and $b$. | $a = 5$, $b = 3$ |
| 18 | A car depreciates at 18% per year. If the new price is CHF 30 000, find the value after 3 years. | $\approx$ CHF 16 549 |
| 19 | Express $\log_a (p^3 q^2) - \log_a(p q^3)$ in terms of $\log_a p$ and $\log_a q$. | $2\log_a p - \log_a q$ |
| 20 | A population is modelled by $P(t) = 500 \cdot (1.04)^t$ where $t$ is years. State the meaning of the values 500 and 1.04. | Initial population 500; growth factor 1.04 per year (4% growth). |
| 21 | CHF 5000 is invested at 4.5% per year, compounded annually. Find the value after 8 years, to the nearest franc. | CHF 7106 |
| 22 | Solve $ 5^x = 17$, giving the answer correct to 3 s.f. | $x \approx 1.76$ |
| 23 | Describe the transformation from $y = 2^x$ to $y = 2^{x - 3} + 1$. | Translate 3 right and 1 up; new asymptote $y = 1$. |
| 24 | Solve $\log_3 x + \log_3(x - 2) = 1$. | $x = 3$ |
| 25 | A model $C(t) = a \cdot b^t$ satisfies $C(1) = 6$ and $C(4) = 162$. Find $a$ and $b$. | $a = 2$, $b = 3$ |
| 26 | How many full years does it take for CHF 1000 to grow to CHF 2000 at 5% per year compounded annually? | 15 years |
| 27 | A population of bacteria triples every 2 hours. If $N(0) = 200$, write a model $N(t) = a \cdot b^t$. | $N(t) = 200 \cdot 3^{t/2}$ or equivalently $200 \cdot (\sqrt{3})^t$ |
| 28 | A model $M(t) = 80 \cdot 0.85^t$ describes the temperature drop of a coffee. Find (a) $M(0)$, (b) $M(10)$ to 1 d.p., (c) the long-run behaviour. | (a) 80°C; (b) $\approx 15.8$°C; (c) tends to 0 as $t \to \infty$. |
| 29 | A function is $y = 3 \cdot 2^{x - 1} + 2$. State (a) the $y$-intercept, (b) the asymptote, (c) whether it grows or decays. | (a) $(0, 3.5)$; (b) $y = 2$; (c) grows |
| 30 | Compare: (a) CHF 1000 at 5% compounded annually for 10 years; (b) CHF 1000 at 4.9% compounded monthly for 10 years. Which gives more, and by how much (to nearest franc)? | (b) is more by $\approx$ CHF 5 (a: 1628.89; b: 1633.99). |
| 31 | Solve $4 \cdot 2^{2x} - 9 \cdot 2^x + 2 = 0$ for real $x$. | $x = -2$ or $x = 1$ |
| 32 | Solve $\log_2 x - \log_2(x - 3) = 2$, giving an exact answer. | $x = 4$ |
| 33 | A population follows $P(t) = 200 \cdot 3^t$ weeks. Find the smallest $t$ (to 3 s.f.) for which $P > 100\,000$. | $t \approx 5.66$ weeks |
| 34 | An investment doubles in 8 years with annual compounding. Find the annual interest rate $r$ (to 3 s.f.). | $r \approx 9.05\%$ |
| 35 | A bacteria colony is modelled by $N(t) = a \cdot e^{kt}$. At $t = 0$, $N = 200$; at $t = 5$, $N = 800$. Find $a$ and $k$ (to 3 s.f.). | $a = 200$; $k \approx 0.277$ |
| 36 | A radioactive isotope has a half-life of 12 years. Find the proportion remaining after 30 years (to 3 d.p.). | $\approx 0.177$ (17.7%) |
| 37 | Evaluate $\log_4 32$ exactly using the change-of-base law. | $\dfrac{5}{2}$ |
| 38 | The graph of $y = a \cdot 2^x + c$ has $y$-intercept $(0, 4)$ and horizontal asymptote $y = -3$. Find $a$ and $c$. | $a = 7$, $c = -3$ |
| 39 | A model $h(t) = h_0 \cdot r^t$ describes the bounce height of a ball. After bounce 1, height is 80 cm; after bounce 4, height is 32.768 cm. Find $h_0$ and $r$. | $h_0 = 100$, $r = 0.8$ |
| 40 | Solve $2^{x + 1} = 5 \cdot 3^x$ to 3 s.f. | $x \approx -2.26$ |
Pack B — Answers
| # | Question | Answer |
|---|---|---|
| 1 | Simplify $x^{ 7} \cdot x^{ 2}$ as a single power of $x$. | $x^{9}$ |
| 2 | Simplify $\dfrac{x^{ 10}}{x^{ 4}}$ as a single power of $x$. | $x^{6}$ |
| 3 | Simplify $\left(z^{ 5}\right)^{ 2}$ as a single power. | $z^{10}$ |
| 4 | Write $\left(\dfrac{1}{2}\right)^{-4}$ as a whole number. | 16 |
| 5 | Evaluate $(1024)^0$. | 1 |
| 6 | State the $y$-intercept and the equation of the horizontal asymptote of $y = 3^x$. | $y$-intercept $(0, 1)$; asymptote $y = 0$ |
| 7 | Evaluate $ 3^{ 4}$. | 81 |
| 8 | Evaluate $ 36^{1/2}$ exactly. | 6 |
| 9 | Solve $ 3^x = 27$, giving an exact answer. | $x = 3$ |
| 10 | CHF\,1000 is invested at 5% per year, compounded annually. Find the value after 1 year. | CHF 2080 |
| 11 | Write $ 3^{ 4} = 81$ in logarithmic form. | $\log_3 81 = 4$ |
| 12 | Evaluate $\log_{ 2} 64$. | 6 |
| 13 | A radioactive substance decays at 12% per year. Find the multiplier per year, and the amount after 5 years if initial mass is 100 g. | Multiplier 0.92; after 4 years $\approx 35.8$ g |
| 14 | Evaluate $ 27^{2/3}$ exactly. | 9 |
| 15 | A colony of 200 bacteria doubles every hour. Find the population after 4 hours. | 4800 |
| 16 | Simplify $\log 8 + \log 5 - \log 4$. | 1 |
| 17 | A function $y = a \cdot b^x$ has $y$-intercept 5 and passes through $(1, 15)$. Find $a$ and $b$. | $a = 4$, $b = 3$ |
| 18 | A car depreciates at 18% per year. If the new price is CHF 30 000, find the value after 3 years. | $\approx$ CHF 20 880 |
| 19 | Express $\log_a (p^3 q^2) - \log_a(p q^3)$ in terms of $\log_a p$ and $\log_a q$. | $-2\log_a p + 3\log_a q$ |
| 20 | A population is modelled by $P(t) = 500 \cdot (1.04)^t$ where $t$ is years. State the meaning of the values 500 and 1.04. | Initial population 200; growth factor 1.08 (8% per year). |
| 21 | CHF 5000 is invested at 4.5% per year, compounded annually. Find the value after 12 years, to the nearest franc. | CHF 8474 |
| 22 | Solve $ 7^x = 50$, giving the answer correct to 3 s.f. | $x \approx 2.01$ |
| 23 | Describe the transformation from $y = 2^x$ to $y = 2^{x - 3} + 1$. | Translate 2 left and 4 down; new asymptote $y = -4$. |
| 24 | Solve $\log_3 x + \log_3(x - 2) = 1$. | $x = 2$ |
| 25 | A model $C(t) = a \cdot b^t$ satisfies $C(1) = 6$ and $C(4) = 162$. Find $a$ and $b$. | $a = 2$, $b = 5$ |
| 26 | How many full years does it take for CHF 1000 to grow to CHF 2000 at 5% per year compounded annually? | 11 years |
| 27 | A population of bacteria triples every 2 hours. If $N(0) = 200$, write a model $N(t) = a \cdot b^t$. | $N(t) = 100 \cdot 2^{t/3}$ |
| 28 | A model $M(t) = 80 \cdot 0.85^t$ describes the temperature drop of a coffee. Find (a) $M(0)$, (b) $M(10)$ to 1 d.p., (c) the long-run behaviour. | (a) 60°C; (b) $\approx 20.9$°C; (c) tends to 0. |
| 29 | A function is $y = 3 \cdot 2^{x - 1} + 2$. State (a) the $y$-intercept, (b) the asymptote, (c) whether it grows or decays. | (a) $(0, 35)$; (b) $y = -1$; (c) grows |
| 30 | Compare: (a) CHF 1000 at 5% compounded annually for 10 years; (b) CHF 1000 at 4.9% compounded monthly for 10 years. Which gives more, and by how much (to nearest franc)? | (b) is more by $\approx$ CHF 12. |
| 31 | Solve $4 \cdot 2^{2x} - 9 \cdot 2^x + 2 = 0$ for real $x$. | $x = 0$ or $x = 1$ |
| 32 | Solve $\log_2 x - \log_2(x - 3) = 2$, giving an exact answer. | $x = 3$ |
| 33 | A population follows $P(t) = 200 \cdot 3^t$ weeks. Find the smallest $t$ (to 3 s.f.) for which $P > 100\,000$. | $t \approx 12.7$ weeks |
| 34 | An investment doubles in 8 years with annual compounding. Find the annual interest rate $r$ (to 3 s.f.). | $r \approx 5.95\%$ |
| 35 | A bacteria colony is modelled by $N(t) = a \cdot e^{kt}$. At $t = 0$, $N = 200$; at $t = 5$, $N = 800$. Find $a$ and $k$ (to 3 s.f.). | $a = 100$; $k \approx 0.693$ |
| 36 | A radioactive isotope has a half-life of 12 years. Find the proportion remaining after 30 years (to 3 d.p.). | $\approx 0.138$ (13.8%) |
| 37 | Evaluate $\log_4 32$ exactly using the change-of-base law. | $\dfrac{5}{3}$ |
| 38 | The graph of $y = a \cdot 2^x + c$ has $y$-intercept $(0, 4)$ and horizontal asymptote $y = -3$. Find $a$ and $c$. | $a = 6$, $c = 1$ |
| 39 | A model $h(t) = h_0 \cdot r^t$ describes the bounce height of a ball. After bounce 1, height is 80 cm; after bounce 4, height is 32.768 cm. Find $h_0$ and $r$. | $h_0 = 125$, $r = 0.8$ |
| 40 | Solve $2^{x + 1} = 5 \cdot 3^x$ to 3 s.f. | $x \approx 1.22$ |
Problems — Worked Solutions
**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) $\left(\dfrac{1}{x^3}\right)^{-2}$
(a) $x^7$. (b) $x^5$. (c) $x^{10}$. (d) $x^6$.
**Fractional indices [EXT].** Evaluate exactly. (a) $25^{1/2}$ (b) $8^{2/3}$ (c) $16^{3/4}$ (d) $\left(\dfrac{1}{27}\right)^{-2/3}$
(a) 5. (b) 4. (c) 8. (d) 9.
**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?
(a) $V = 5000(1.045)^t$. (b) CHF 7106. (c) $t = 16$. (d) $\approx 200\%$ (tripled).
**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).
(a) $a = 2$, $b = 3$. (b) $C(0) = 2$, i.e. 200 customers. (c) Increasing exponential through $(0, 2)$ and $(4, 162)$. (d) $t \approx 5.66$ weeks.
**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 $y = 2^{x - 3} + 1$.
(a) Domain $\mathbb{R}$, range $y > 0$, $y$-intercept $(0, 1)$, asymptote $y = 0$. (b) Second graph is a 3-right, 1-up shift of the first. (c) $y$-intercept $\left(0, \frac{9}{8}\right)$; asymptote $y = 1$.
**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.
(a) $k = (0.5)^{1/8}$. (b) $\dfrac{1}{8}$ (i.e. 12.5%). (c) $t \approx 26.6$ years.
**Logarithm equation — extraneous root [EXT].** Solve $\log_3 x + \log_3(x - 2) = 1$ and explain why one algebraic candidate must be rejected.
$x = 3$.
**Hidden quadratic [EXT].** Solve $4^x - 5 \cdot 2^x + 4 = 0$ for real $x$.
$x = 0$ or $x = 2$.
**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 $P(8)$.
$a = 2$, $b = 3$; $P(8) = 13\,122$.
**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).
(a) $V(t) = 32000 \cdot 0.82^t$. (b) CHF $\approx 12\,388$. (c) Year 6.
**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) $\dfrac{1}{2} \log m + \log n$
(a) $\log\dfrac{ab}{c}$. (b) $\log\dfrac{p^2}{q^3}$. (c) $\log(n \sqrt{m})$.
**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.)?
(a) CHF 1060. (b) $\approx$ CHF 1061.68. (c) EAR $\approx 6.17\%$. (d) $\approx 11.6$ years.