F: Exponentials and logarithms

Overview

AQA A-level Mathematics 7357, section 3.7F makes exponentials and logarithms a connected topic rather than a rule-collection. Exponential functions model quantities that change by a constant proportion, such as continuously compounded savings, radioactive material and drug concentration. Logarithms reverse an exponential process. That inverse relationship is why logarithms solve equations with an unknown power, straighten curved data and produce graphs with a recognisable asymptote.

AQA can assess this material in any of the three A-level papers. All papers are 2 hours and 100 marks, so the average working rate is 1.2 minutes per mark. Expect short algebraic manipulations, graph sketches, differentiation, parameter estimation from transformed graphs and modelling questions with a final interpretation. Strong candidates do not just press ln on a calculator. They state restrictions, show equivalent algebraic steps, keep accuracy until the end and explain what a parameter means in context.

The topic connects directly to Algebra and functions, Differentiation, Numerical methods and mathematical modelling. Its central idea is simple: an exponential input and a logarithmic output undo one another. Once candidates see that relationship, the graphs, laws and equation-solving methods become predictable.

Specification map: F1 to F7

AQA requires candidates to know and use y = a^x and y = e^x; understand why the gradient of e^(kx) is k e^(kx); use logarithms as inverse functions; apply the logarithm laws; solve equations of the form a^x = b; linearise exponential and power relationships; and model exponential growth and decay. The specification expressly includes the limitations and refinements of exponential models, not just calculation.

Key concepts

1. Exponential functions and their graphs

For y = a^x, the base must satisfy a > 0 and a ≠ 1. When a > 1, each step to the right multiplies y by a number greater than 1, so the graph grows. When 0 < a < 1, each step multiplies by a fraction, so the graph decays. Every exponential graph goes through (0, 1) because a^0 = 1. Its output is always positive, so it cannot touch or cross the x-axis. The x-axis, y = 0, is therefore a horizontal asymptote.

The natural exponential e^x is the most important version. Its base is Euler's number, e ≈ 2.71828. It appears whenever a quantity changes continuously at a rate proportional to its present size. A graph sketch needs its key point and its asymptote, not a decorative curve. If a question asks for y = e^(x-2) + 3, translate both features: (0,1) becomes (2,3) and y = 0 becomes y = 3.

Graph checklist for marks: label axes; give the intercept or a known point; draw the asymptote as a dashed line where helpful; state a domain or range only when asked; and move the asymptote whenever the graph is translated.

2. Natural logarithms are inverse functions

log_a x means “the power to which a must be raised to obtain x”. For example, log_2 32 = 5 because 2^5 = 32. The natural logarithm, ln x, is log base e. It is the inverse of e^x, so:

• ln(e^x) = x for every real x.
• e^(ln x) = x only when x > 0.

Inverse functions swap input and output. Graphically, y = ln x is the reflection of y = e^x in the line y = x. The point (0,1) on e^x therefore becomes (1,0) on ln x. The exponential horizontal asymptote y = 0 becomes the logarithm vertical asymptote x = 0.

The domain restriction is not optional wording. In real-valued A-level Mathematics, ln x is defined only for x > 0. This is why ln(-3) is not an acceptable real answer and why a model such as y = ln(x-4) has domain x > 4. Credit is given for rejecting extraneous values created by algebra where the original expression is undefined.

3. Logarithm laws: derive, do not guess

The laws of logarithms are index laws written in reverse. Suppose u = log_a x and v = log_a y. Then x = a^u and y = a^v. Multiplying gives xy = a^(u+v), hence log_a(xy) = u+v. Similar reasoning gives the quotient and power laws.

• log_a(xy) = log_a x + log_a y
• log_a(x/y) = log_a x − log_a y
• log_a(x^n) = n log_a x

All arguments must be positive, and a > 0, a ≠ 1. The most common error is inventing a rule for addition: log_a(x+y) is not log_a x + log_a y. Test it with base 10: log(2+3) = log 5, whereas log 2 + log 3 = log 6. They are different.

Memory hook: Products add, quotients subtract, powers come down, but sums stay put. Say it before manipulating an expression. It prevents the most expensive one-line mistake in this topic.

4. Solving exponential and logarithmic equations

When the unknown is in the power, isolate the exponential first and then take logs of both sides. For 5^(2x-1) = 18, natural logs give (2x-1) ln 5 = ln 18, so x = [1 + ln 18 / ln 5] / 2. Use ln consistently, or use log consistently. Mixing bases is still valid only if each side uses the same base at that step.

Show the logarithm step. An answer from a calculator without working risks losing method marks. Retain full precision until the final answer, then round only as requested. If a question gives ln(x-1) = 2, exponentiate to obtain x-1 = e^2 before finding x; the domain check x > 1 is then automatically satisfied.

5. Differentiation and proportional-rate models

d/dx(e^x) = e^x. By the chain rule, d/dx(e^(kx)) = k e^(kx). If y = Ae^(kt), then dy/dt = A k e^(kt) = ky. The rate of change is proportional to the amount. A positive k describes growth; a negative k describes decay.

For a context model, interpret A as the value at t = 0 and give k with its time-unit meaning. For instance, in C = 40e^(-0.3t), C starts at 40 and its instantaneous rate is -0.3C per time unit. The negative sign shows decay. This does not mean that exactly 30% is removed at the end of each unit. It means the continuous proportional rate is 0.3 per unit.

6. Linearising data with logarithms

AQA expects candidates to use logarithmic graphs to estimate parameters. For the exponential relationship y = ab^x, take natural logs:

ln y = ln a + x ln b.

Plot ln y vertically against x horizontally. A straight line has gradient ln b and intercept ln a. Therefore b = e^(gradient) and a = e^(intercept).

For the power relationship y = ax^n, take natural logs:

ln y = ln a + n ln x.

Plot ln y against ln x. Its gradient is n; its intercept is ln a. A frequent error is to report the gradient as b in the first model or a in either model. Translate it back to the parameter the question asks for.

7. Model criticism earns modelling marks

Exponential models assume a constant proportional rate. That can be reasonable over a limited interval, but it can fail. A population may be restricted by food, space or disease. A drug may have changing absorption and elimination rates. Financial growth can vary as the interest rate changes. State a specific reason linked to the context, rather than writing only “the model is unrealistic”. A refinement may be a piecewise model, a changing rate, or a logistic model where a limiting value is plausible.

Formulae and relationships

AQA Appendix B says these logarithm laws and core derivatives must be used without being provided, except where a formula is the starting point or result of a proof. Do not rely on a formula sheet for this topic.

Practical applications

Exponential decay is used in drug concentration. If a dose is repeatedly reduced by the same proportion in equal time intervals, an exponential model may fit early data. Continuous compound interest uses A = Pe^(rt), where P is the initial investment, r is the continuous rate and t is time. Radioactive decay and population growth can use the same mathematical form, although the model is only defensible while its assumptions are reasonable.

There is no required practical for AQA 7357 section 3.7F. The assessed practical-style skill is data handling: transform data, produce a correctly labelled graph, identify gradient and intercept, estimate original parameters, and evaluate the model. No unit conversions are intrinsic to the topic, but time units must be consistent with the rate constant in a model.

Active recall before moving on

Cover the formula table. Sketch e^x and ln x with their key points and asymptotes. Then say the three log laws aloud and explain why a logarithm cannot accept a non-positive argument. If you cannot do those four things unaided, return to the worked examples rather than reading the notes again.

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