CAIE A-Level Mathematics 2.2 Logarithmic and Exponential Functions Question Bank

Practise converting between logarithmic and exponential forms, solving equations and inequalities and linearising exponential models to interpret gradients and intercepts.

Exam points

combine logarithms with product, quotient and power laws before removing them
take logarithms of exponential equations or inequalities and preserve the inequality direction
linearise an exponential model and use the straight-line gradient and intercept to find constants

Covered syllabus points

2.2.1—Log laws 2.2.2—Exponential and ln 2.2.3—Log equations 2.2.4—Linearising relationships

[Maximum number: 3]

$5 \quad$ (a) Given that $2 \ln (x+1)+\ln x=\ln (x+9)$ , show that $x=\sqrt{\frac{9}{x+2}}$ .

Use the power law correctly
*M1
Use correct process to obtain equation with no logarithms
DM1
Confirm $x=\sqrt{\frac{9}{x+2}}$
A1
AG; condone absence of justification for choice of positive root.

[Maximum number: 3]

The polynomial p(x) is defined by

\mathrm{p}(x)=2 x^{3}+a x^{2}-3 x-4

where a is a constant. It is given that ( x-4 ) is a factor of p(x).

Show that the equation $\mathrm{p}\left(\mathrm{e}^{3 y}\right)=0$ has only one real root and find its exact value.

Apply logarithms and use power law for $\mathrm{e}^{3 y}=4$

M1

Obtain $\frac{1}{3} \ln 4$ or exact equivalent

A1

Use discriminant [=1-8=-7] or equivalent to show no other roots

B1

AG - necessary detail needed

[Maximum number: 2]

Hence solve the equation $\mathrm{e}^{-12 y}-32 \mathrm{e}^{-3 y}+48=0$ , giving your answer in an exact form.

Apply logarithms and use power law for $\mathrm{e}^{-3 y}=k$ where k>0
M1
Obtain $y=-\frac{1}{3} \ln 2, \frac{1}{3} \ln \frac{1}{2}$
A1
Or exact equivalent
Must be simplified e.g. not lne or $\frac{6}{3}$
ISW extra solutions but A0 if undefined solutions are included.

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Covered syllabus points

2.2 Logarithmic and exponential functions question 1

2.2 Logarithmic and exponential functions question 2

2.2 Logarithmic and exponential functions question 3