Pack B — Fluency

Mathematics

Year 11 · 11.6 Exponential and Logarithmic Functions

Pack B · Fluency

Name: _________________________________

Date: _________________ Class: ___________

Answer all questions. Show your working. Questions are grouped by challenge level.

BronzeQuestions 1–10

1.
Simplify $x^{ {a}} \cdot x^{ {b}}$ as a single power of $x$ .
2.
Simplify $\dfrac{x^{ {a}}}{x^{ {b}}}$ as a single power of $x$ .
3.
Simplify $\left(z^{ {a}}\right)^{ {b}}$ as a single power.
4.
Write $\left(\dfrac{1}{2}\right)^{-{n}}$ as a whole number.
5.
Evaluate $(1024)^0$ .
6.
State the $y$ -intercept and the equation of the horizontal asymptote of $y = 3^x$ .
7.
Evaluate $3^{ {n}}$ .
8.
Evaluate $36^{1/2}$ exactly.
9.
Solve $3^x = 27$ , giving an exact answer.
10.
CHF\,1000 is invested at 5% per year, compounded annually. Find the value after 1 year.

SilverQuestions 11–20

11.
Write $3^{ {x}} = 81$ in logarithmic form.
12.
Evaluate $\log_{ {a}} 64$ .
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.
14.
Evaluate $27^{2/3}$ exactly.
15.
A colony of 200 bacteria doubles every hour. Find the population after 4 hours.
16.
Simplify $\log 8 + \log 5 - \log 4$ .
17.
A function $y = a \cdot b^x$ has $y$ -intercept 5 and passes through $(1, 15)$ . Find $a$ and $b$ .
18.
A car depreciates at 18% per year. If the new price is CHF 30 000, find the value after 3 years.
19.
Express $\log_a (p^3 q^2) - \log_a(p q^3)$ in terms of $\log_a p$ and $\log_a q$ .
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.

GoldQuestions 21–30

21.
CHF 5000 is invested at 4.5% per year, compounded annually. Find the value after 12 years, to the nearest franc.
22.
Solve $7^x = 50$ , giving the answer correct to 3 s.f.
23.
Describe the transformation from $y = 2^x$ to $y = 2^{x - 3} + 1$ .
24.
Solve $\log_3 x + \log_3(x - 2) = 1$ .
25.
A model $C(t) = a \cdot b^t$ satisfies $C(1) = 6$ and $C(4) = 162$ . Find $a$ and $b$ .
26.
How many full years does it take for CHF 1000 to grow to CHF 2000 at 5% per year compounded annually?
27.
A population of bacteria triples every 2 hours. If $N(0) = 200$ , write a model $N(t) = a \cdot b^t$ .
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.
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.
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)?

PlatinumQuestions 31–40

31.
Solve $4 \cdot 2^{2x} - 9 \cdot 2^x + 2 = 0$ for real $x$ .
32.
Solve $\log_2 x - \log_2(x - 3) = 2$ , giving an exact answer.
33.
A population follows $P(t) = 200 \cdot 3^t$ weeks. Find the smallest $t$ (to 3 s.f.) for which $P > 100\,000$ .
34.
An investment doubles in 8 years with annual compounding. Find the annual interest rate $r$ (to 3 s.f.).
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.).
36.
A radioactive isotope has a half-life of 12 years. Find the proportion remaining after 30 years (to 3 d.p.).
37.
Evaluate $\log_4 32$ exactly using the change-of-base law.
38.
The graph of $y = a \cdot 2^x + c$ has $y$ -intercept $(0, 4)$ and horizontal asymptote $y = -3$ . Find $a$ and $c$ .
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 $r$ .
40.
Solve $2^{x + 1} = 5 \cdot 3^x$ to 3 s.f.