Edexcel A-Level Mathematics A2 P3.3 Exponentials and Logarithms Questions

Practise exponential and logarithmic models, graphs and transformations, including asymptotes, inverse links, and linear log relationships.

Exam points

sketch exponential or logarithmic curves and state asymptotes or limiting values
use ln to solve exponential model equations without relying only on calculator technology

Covered syllabus points

P3.3.1 - function ex and its graph P3.3.2 - function ln x and its graph P3.3.3 - Use logarithmic graphs to estimate

[Maximum number: 5]

A scientist is studying a population of fish in a lake. The number of fish, N, in the population, t years after the start of the study, is modelled by the equation

N=\frac{600 \mathrm{e}^{0.3 t}}{2+\mathrm{e}^{0.3 t}} \quad t \geqslant 0

Use the equation of the model to answer parts (a), (b), (c), (d) and (e).

(a)

Find the number of fish in the lake at the start of the study.

[ 1 ]

(b)

Find the time, after the start of the study, when there are predicted to be 500 fish in the lake. Give your answer in years and months to the nearest month.

[ 4 ]

$500=\frac{600e^{0.3t}}{2+e^{0.3t}}\Rightarrow 100e^{0.3t}=1000\Rightarrow e^{0.3t}=10$ $t=\frac{\ln 10}{0.3}=7\text{ years }8\text{ months}$

[Maximum number: 3]

Figure 1

The line l in Figure 1 shows a linear relationship between $\log _{10} y$ and x.
The line passes through the points (0,1.5) and (-4.8,0) as shown.

Hence, or otherwise, express y in the form $k b^{x}$ , giving the values of the constants k and b to 3 significant figures.

$\log_{10}y=\frac{5}{16}x+1.5\Rightarrow y=10^{\frac{5}{16}x+1.5}$ $y=10^{\frac{5}{16}x}\times 10^{1.5}$ $y=31.6\times 2.05^x$
(3)

M1: Removes logs from $\log_{10}y=mx+c$ .

M1: Correct strategy to obtain k and b.

A1: Correct equation $y=31.6\times 2.05^x$ , no errors seen.

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Covered syllabus points

Question 1

Question (a)

Question (b)

Question 2