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Exponential and Logarithmic Functions · Paper 1 — Free Maths AA HL Practice Questions

1FoundationSAQ-SExponent laws and propertiess5 marksPaper 1~8 min

The number of bacteria in a culture is given by

N(t) = 250 \times 3^{kt}

, where

t

is the time in hours after the culture is first observed. After

2

hours, the number of bacteria is

4500

(a)

Calculate the value of

k

, giving your answer in the form

k = \log_3 a

where

a \in \mathbb{Q}

. [2 marks]

(b)

Calculate the number of bacteria present after

5

hours, giving your answer to the nearest integer. [3 marks]

Solutions

2MasterySAQ-SExponent laws and propertiess5 marksPaper 1~8 min

The number of bacteria in a culture,

N(t)

, after

t

hours is modelled by

N(t) = N_0 \cdot 2^{kt}

, where

N_0

is the initial number of bacteria and

k

is a positive constant.

(a)

Show that when

k = 1

, the population doubles every hour. [2 marks]

(b)

Hence, find the value of

k

such that the population triples every 4 hours. Give your answer in the form

\log_2 a

where

a \in \mathbb{Q}^+

. [3 marks]

Solutions

3ChallengeSAQ-LLogarithmic functions and their propertiess8 marksPaper 1~12 min

The graph of

y = \ln(ax + b)

passes through the points

P(1,\, \ln 5)

and

Q(3,\, \ln 13)

, where

a

and

b

are real constants with

ax + b > 0

for all

x

in the domain.

(a)

Determine the exact values of

a

and

b

. [4 marks]

(b)

Hence, find the exact coordinates of the point where the graph intersects the

x

-axis. [2 marks]

(c)

Show that

y = \ln(ax + b)

can be written in the form

y = \ln a + \ln\!\left(x + \dfrac{b}{a}\right)

, and hence describe fully the two transformations that map the graph of

y = \ln x

onto the graph of

y = \ln(ax + b)

, using your values of

a

and

b

. [2 marks]

Solutions

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