CAIE A-Level Mathematics A2 3.2 Logarithmic and Exponential Functions Questions

Practise solving logarithmic and exponential equations and transforming growth or decay relationships into straight-line form to calculate model constants and times.

Exam points

apply product, quotient and power laws before isolating the unknown in a log equation
take logarithms of exponential relations and solve the resulting linear equation accurately
match a transformed straight line to Y = mX + c and recover original model constants

Covered syllabus points

3.2.1—Log laws 3.2.2—Exponential and ln 3.2.3—Log equations 3.2.4—Linearising relationships

[Maximum number: 4]

Solve the equation $\ln (2 x-1)=2 \ln (x+1)-\ln x$ . Give your answer correct to 3 decimal places.

Use law for the logarithm of a product, quotient or power
Remove logarithms and state a correct equation, e.g. $x(2 x-1)=(x+1)^{2}$
Solve a 3-term quadratic obtaining at least one root
Obtain answer 3.303 only

[Maximum number: 4]

Solve the equation $4^{x-2}=4^{x}-4^{2}$ , giving your answer correct to 3 decimal places.

Use laws of indices correctly and solve for $4^{x}$
Obtain correct solution in any form, e.g. $4^{x}=\frac{256}{15}$
Use a correct method for solving an equation of the form $4^{x}=a$ , where
a>0
Obtain answer 2.047

[Maximum number: 4]

Figure for Question 3 — CAIE A-Level Mathematics A2

The variables x and y are related by the equation $y=a b^{x}$ , where a and b are constants. The diagram shows the result of plotting $\ln y$ against x for two pairs of values of x and y. The coordinates of these points are ( 1,3.7 ) and ( 2.2,6.46 ).

Use this information to find the values of a and b.

State or imply that $\ln y=\ln a+x \ln b$
Carry out a completely correct method for finding $\ln a$ or $\ln b$ $3.7=\ln a+\ln b$ and $6.46=\ln a+2.2 \ln b$
leading to $\ln a=1.4, \ln b=2.3$ .

Obtain value a=4.06
Obtain value b=9.97
SC B1 for $a=\mathrm{e}^{1.4}$ and $b=\mathrm{e}^{2.3}$ .

Alternative Method for Question 3

$e^{3.7}=a b^{1}$ and $e^{6.46}=a b^{2.2}$
Divide to obtain $\mathrm{e}^{2.76}=b^{1.2}$ and state or imply $2.76=1.2 \ln b$
Obtain value a=4.06
Obtain value b=9.97

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Covered syllabus points

Question 1

Question 2

Question 3