Senior Secondary (HKDSE) · Mathematics M1 (Calculus and Statistics)

Exponential and logarithmic functions: Practice Questions

5 multiple-choice questions marked as you go, and 5 written questions with worked solutions. All on Exponential and logarithmic functions.

10 questions25 marksFree, no account

Question 1

1 mark

Evaluate \( \log_2 (64) + \log_{10} (0.001) \).

Question 2

1 mark

Find the derivative of $$f(x) = \frac{\ln x}{x}$$.

Question 3

1 mark

The variables \(x\) and \(y\) satisfy the relation \(y = \frac{k}{(x^2+1)^n}\), where \(k\) and \(n\) are positive constants. It is given that the graph of \(\ln y\) against \(\ln(x^2+1)\) is a straight line passing through the points \( (1, 5) \) and \( (4, -1) \). Find the value of \(n\).

Question 4

1 mark

Simplify the expression \( \frac{\log_4 16}{\log_2 8} + \log_5 (\frac{1}{25}) \).

Question 5

1 mark

A population of bacteria \(N\) grows according to the model \(N = N_0 e^{kt}\), where \(t\) is the time in hours and \(N_0, k\) are constants. If the population triples every 5 hours, find the value of \(k\) correct to 3 decimal places.

Question 6

2 marks

If \( \ln(x^2) = 6 \) and \( x > 0 \), find the exact value of \( x \) in terms of the constant \( e \).

Write your answer out first, then check it against the worked solution.

Question 7

3 marks

Using the first three terms of the exponential series \( e^x \approx 1 + x + \frac{x^2}{2!} \), find an approximate value of \( e^{0.1} + e^{-0.1} \).

Write your answer out first, then check it against the worked solution.

Question 8

5 marks

The variables \( x \) and \( y \) are related by the equation \( y = Ax^n \). When \( \ln y \) is plotted against \( \ln x \), a straight line is obtained which passes through the points \( (2, 5) \) and \( (4, 1) \). Determine the value of \( n \) and express the constant \( A \) in terms of \( e \).

Write your answer out first, then check it against the worked solution.

Question 9

4 marks

A certain radioactive substance decays such that its mass \(m\) (in grams) at time \(t\) (in years) is given by the model \(m = 100(0.95^t)\).
(a) Find the initial mass of the substance when \(t = 0\).
(b) Find the time required for the mass to decay to half of its initial value. Give your answer correct to 2 decimal places.

Write your answer out first, then check it against the worked solution.

Question 10

6 marks

The variables \(x\) and \(y\) satisfy the relation \(y = kx^n\), where \(k\) and \(n\) are positive constants. The graph of \(\ln y\) against \(\ln x\) is a straight line as shown in the figure. The line passes through the points \(A(2, 5)\) and \(B(4, 9)\).
(a) Express \(\ln y\) as a linear function of \(\ln x\).
(b) Find the values of \(n\) and \(\ln k\). Hence, find the value of \(k\) in terms of \(e\).

Write your answer out first, then check it against the worked solution.

* The content provided by thinka is generated by AI and may not always be accurate or up-to-date. Please use it as a supplementary resource and verify with official materials.

You've seen the model answer. Now get yours marked.

This page can show you how a good answer looks. It cannot tell you what your answer was missing. thinka marks your written work against the real mark scheme in about 15 seconds.

Get My Answers Marked Back to the notes

Want more questions like these? Get a fresh set on this topic, graded as you go.