Problem-solving Pack

Mathematics

Year 11 · 11.6 Exponential and Logarithmic Functions

Problem-solving Pack

Name: _________________________________

Date: _________________ Class: ___________

These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.

1Problem 1 of 12

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}

Working space

2Problem 2 of 12

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}

Working space

3Problem 3 of 12

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?

Working space

4Problem 4 of 12

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).

Working space

5Problem 5 of 12

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

Working space

6Problem 6 of 12

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.

Working space

7Problem 7 of 12

Logarithm equation — extraneous root [EXT]. Solve

\log_3 x + \log_3(x - 2) = 1

and explain why one algebraic candidate must be rejected.

Working space

8Problem 8 of 12

Hidden quadratic [EXT]. Solve

4^x - 5 \cdot 2^x + 4 = 0

for real

x

Working space

9Problem 9 of 12

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)

Working space

10Problem 10 of 12

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).

Working space

11Problem 11 of 12

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

Working space

12Problem 12 of 12

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.)?

Working space