O-Level Additional Mathematics: Exponential & Logarithmic Functions — Turning Curves into Straight Lines to Score Full Marks

Why this topic matters for the exam

Exponential and logarithmic functions appear in almost every O-Level Additional Mathematics paper, and they are generous with marks if you are disciplined. Examiners love this topic because it tests several skills at once: manipulating indices, applying the laws of logarithms, solving equations carefully, and reading information off a graph. The most prized question of all is the "convert to linear form" question, where a non-linear relationship is turned into a straight line so that unknown constants can be read off the gradient and intercept. Students who master this technique often pick up an easy 6 to 8 marks while their classmates lose them.

The good news is that this is a learnable topic. There is no guessing involved. Once a student knows the laws, recognises the question types, and builds the habit of checking solutions, the marks become very predictable. Let us build that foundation properly.

Core concepts clearly explained

1. A quick recap of the index laws

Everything in this topic rests on the laws of indices. Make sure these are second nature:

a^m × aⁿ = a^m+n
a^m ÷ aⁿ = a^m−n
(a^m)ⁿ = a^mn
a⁰ = 1 (for any a ≠ 0)
a⁻ⁿ = 1 ÷ aⁿ
a^1/n = the n-th root of a, so a^1/2 = √a

A common early trap: a^m + aⁿ does not simplify to a^m+n. The index laws only combine terms that are multiplied or divided, never added.

2. The exponential function

An exponential function has the form y = a^x, where the base a is a positive constant and the variable x sits in the power. For example, y = 2^x grows quickly: when x increases by 1, y doubles. The graph of y = a^x (for a > 1) always passes through (0, 1), rises steeply to the right, and gets closer and closer to the x-axis on the left without ever touching it. That x-axis is a horizontal asymptote.

A special base appears so often that it has its own symbol: e, where e ≈ 2.718. The function y = e^x is called the exponential function. It behaves like other exponentials, passing through (0, 1) and having the x-axis as an asymptote, but e^x has special calculus properties you will meet later. For now, treat e as a fixed number that lives on your calculator.

3. Logarithms as the inverse of exponentials

A logarithm answers the question: "To what power must I raise the base to get this number?" The statement

log_a y = x means exactly the same thing as a^x = y.

So log₂ 8 = 3 because 2³ = 8. Because the logarithm undoes the exponential, the graph of y = log_a x is the reflection of y = a^x in the line y = x.

Two logarithms have shorthand names you must know for the O-Level:

lg means log to base 10 (the common logarithm). So lg x = log₁₀ x.
ln means log to base e (the natural logarithm). So ln x = log_e x.

4. The laws of logarithms

These three laws are the engine of the whole topic. For a positive base a (with a ≠ 1) and positive values of x and y:

Product law: log_a (xy) = log_a x + log_a y
Quotient law: log_a (x ÷ y) = log_a x − log_a y
Power law: log_a (xⁿ) = n log_a x

Two useful special values fall straight out of the definition: log_a 1 = 0 (because a⁰ = 1) and log_a a = 1 (because a¹ = a).

5. Change of base

Your calculator only has lg and ln buttons, but questions may use other bases. The change of base formula lets you convert any logarithm:

log_a b = (log_c b) ÷ (log_c a)

In practice you usually change to base 10 or base e, for example log₂ 5 = (lg 5) ÷ (lg 2) ≈ 2.32. Change of base is also handy for solving equations where the unknown is buried inside the base or where two logs have different bases.

6. The key graph features

For y = a^x with a > 1: domain is all real x; range is y > 0; passes through (0, 1); horizontal asymptote at y = 0 (the x-axis); always increasing.

For y = log_a x with a > 1: domain is x > 0 only; range is all real y; passes through (1, 0); vertical asymptote at x = 0 (the y-axis); always increasing. Notice how the domain and range swap compared with the exponential — that is the inverse relationship in action. The fact that the logarithm only exists for x > 0 is the source of one of the most common exam traps, as we will see.

Worked exam-style examples

Example 1 — Solving an exponential equation using logarithms

Solve 3^2x+1 = 20, giving your answer correct to 3 significant figures.

The unknown is in the power, so we take logarithms of both sides. Either lg or ln works; we will use lg.

Take lg of both sides: lg (3^2x+1) = lg 20.
Use the power law to bring the exponent down: (2x + 1) lg 3 = lg 20.
Divide both sides by lg 3: 2x + 1 = (lg 20) ÷ (lg 3) = 1.30103 ÷ 0.47712 = 2.72683.
Subtract 1: 2x = 1.72683, so x = 0.86342.

Answer: x = 0.863 (3 s.f.). Always carry extra decimal places through the working and only round at the very end.

Example 2 — A hidden quadratic in disguise

Solve 4^x − 5(2^x) + 4 = 0.

This looks frightening until you notice that 4^x = (2²)^x = (2^x)². Let u = 2^x. The equation becomes a familiar quadratic:

u² − 5u + 4 = 0.
Factorise: (u − 1)(u − 4) = 0, so u = 1 or u = 4.
Replace u with 2^x. If 2^x = 1, then x = 0 (since 2⁰ = 1). If 2^x = 4, then x = 2 (since 2² = 4).

Answer: x = 0 or x = 2. Note that a substitution like this can never give a negative value of u, because 2^x is always positive — if a quadratic in u gave a negative root, you would simply reject it.

Example 3 — A logarithmic equation where you must reject an invalid solution

Solve log₃ (x + 6) + log₃ (x − 2) = 2.

Combine the left side using the product law: log₃ [(x + 6)(x − 2)] = 2.
Rewrite in index form: (x + 6)(x − 2) = 3² = 9.
Expand: x² + 4x − 12 = 9, so x² + 4x − 21 = 0.
Factorise: (x + 7)(x − 3) = 0, so x = −7 or x = 3.
Check both in the original equation. If x = −7, then x + 6 = −1, and log₃ (−1) does not exist — you cannot take the log of a negative number. So x = −7 must be rejected. If x = 3, then x + 6 = 9 > 0 and x − 2 = 1 > 0, both valid.

Answer: x = 3 only. The marks for rejecting x = −7 with a reason are marks many students throw away.

Example 4 — Converting to linear form to find constants from a graph

Two variables x and y are related by y = k xⁿ, where k and n are constants. When experimental values of lg y are plotted against lg x, a straight line is obtained passing through the points (1, 5) and (3, 1). Find the value of n and of k.

This is the classic question. The relationship y = k xⁿ is a curve, but we can straighten it by taking lg of both sides:

Take lg of both sides: lg y = lg (k xⁿ).
Apply the product law: lg y = lg k + lg (xⁿ).
Apply the power law: lg y = n lg x + lg k.

Now compare this with the straight-line form Y = mX + c, where Y = lg y and X = lg x. The gradient m = n, and the vertical intercept c = lg k.

Find the gradient from the two given points: m = (1 − 5) ÷ (3 − 1) = −4 ÷ 2 = −2. So n = −2.

Find the intercept c by substituting one point into Y = −2X + c. Using (1, 5): 5 = −2(1) + c, so c = 7. Then lg k = 7, which gives k = 10⁷.

The same method works for the other common model, y = k a^x. Taking lg of both sides gives lg y = (lg a) x + lg k, so this time you plot lg y against x (not against lg x). The gradient is lg a and the intercept is lg k. Deciding what to plot is half the question: if the unknown is a power of x, take lg of both sides and plot lg y against lg x; if the unknown is the base of an exponential, plot lg y against x directly.

Common exam traps and mistakes

Taking the log of a negative or zero. log_a x only exists for x > 0. After solving a logarithmic equation, always substitute back and reject any solution that makes any logged expression negative or zero. State the rejection clearly.
Misapplying the log laws. log (x + y) is not log x + log y. The product law applies to log (xy), a multiplication inside the bracket. Similarly, (log x)² is not the same as log (x²); the power law needs the power inside the logarithm.
Forgetting the power law when solving exponentials. After taking logs of a^x, you must bring the power down to get x lg a. Students often write lg (a^x) = lg a and lose the variable.
Plotting the wrong variables in linear-form questions. Read the question to see whether the unknown sits as a power of x (plot lg y vs lg x) or as the base of an exponential (plot lg y vs x). Choosing wrongly makes the points fail to lie on a line.
Confusing the intercept with the constant. In y = k a^x, the intercept gives lg k, not k itself. You must undo the logarithm: k = 10^(intercept), or k = e^(intercept) if you used ln.
Rounding too early. Keep full calculator accuracy through the working and round only the final answer to the required significant figures.
Mixing bases. If a question uses ln, stay in ln throughout; if it uses lg, stay in lg. Only switch deliberately with the change-of-base formula.

Modelling and applications

Exponential and logarithmic models describe many real situations: the growth of savings under compound interest, the cooling of a hot drink, radioactive decay, and population growth. A typical applied question might state that the temperature T of a liquid after t minutes is T = 80 e^−0.05t, and ask for the temperature after 10 minutes or the time taken to cool to a certain value. The first part is direct substitution; the second part requires you to take ln of both sides to free the variable from the power — exactly the skill from Example 1. Recognising the same techniques in a wordy context is what separates a B from an A.

How to practise and revise

Memorise the laws cold. Write out the three log laws, change of base, and the meanings of lg and ln from memory until you never hesitate.
Drill each question type separately first. Do a set of "solve the exponential equation" questions, then a set of "solve the log equation and reject" questions, then a set of "linear form" questions. Mixing them too early hides which technique you are weak on.
Always check log solutions. Build the habit of substituting back. Make the rejection of invalid solutions automatic.
Practise sketching both graphs y = a^x and y = log_a x, labelling the intercept and the asymptote each time.
Work through past O-Level and prelim papers under timed conditions, and mark your own work against the solutions so you learn where marks are awarded — especially the method marks in linear-form questions.

With the laws secure and the question types familiar, exponentials and logarithms become one of the most dependable scoring topics in A-Math, and if your child would like structured guidance and worked practice, our tutors at Intuitional are always happy to help.