A Level · Cambridge (CIE) · 9709

Solve exponential equations using logarithms and positive substitutions

Bring an unknown exponent into a linear equation, or use a positive exponential variable to reduce an equation to a quadratic.

Section 2.2 · Logarithmic and exponential functions

Pure Mathematics 2 & 3 · Logarithmic and exponential functions · 2.2 / 3.2

Exam scope and assessment notes

Pure Mathematics 2 (Paper 2, AS only) and Pure Mathematics 3 (Paper 3, A Level) · Mathematics 9709, 2026–2027. Paper 1 knowledge is assumed. The AS route using Papers 1 + 2 cannot be carried forward to A Level. Shared sections 2.2 and 3.2 include log laws excluding change of base, natural exponential and logarithmic graphs, equations and inequalities, and straight-line transformations. Written practice is self-checked.

Before you begin

Apply the power law for logarithms and solve linear equations.
Factor quadratics and recognise a positive exponential output.

Work on paper. Try each question before revealing help, and explain why your method works.

Quick readiness check

Taking ln of 3^(x + 1) gives which expression?

What you will learn

Use the logarithmic power law to solve equations with an unknown in the exponent.
Check the positive domain of an exponential substitution and preserve all valid roots.

Taking a logarithm exposes the exponent

For a positive base a and positive target b, an equation a^(g(x)) = b can be solved by taking ln of both sides: g(x) ln a = ln b. This uses the log power law directly. Keep the exact logarithms while solving, then round only if requested. For example 2ˣ = 5 gives x ln 2 = ln 5 and x = (ln 5)/(ln 2), approximately 2.32. Check the result in the original exponential equation.

For 3^(x + 1) = 4^(2x − 1), taking ln gives (x + 1)ln 3 = (2x − 1)ln 4. Collect x terms to get x(2 ln 4 − ln 3) = ln 3 + ln 4 = ln 12. Therefore x = ln 12/(2 ln 4 − ln 3), approximately 1.48. The coefficient being divided by is non-zero; if such a coefficient were zero, inspect whether the remaining equation is always true or impossible rather than divide by zero.

A substitution must retain its positive range

If e^(2x) and eˣ occur together, set u = eˣ > 0 so e^(2x) = u². For e^(2x) − 5eˣ + 6 = 0, the quadratic factors as (u − 2)(u − 3), giving both x = ln 2 and x = ln 3. A positive substitution can still give two valid values; do not stop at the first factor. A negative or zero quadratic candidate is not in the range of eˣ and cannot be turned into a real logarithm.

For e^(2x) + eˣ − 6 = 0, the quadratic is (u + 3)(u − 2) = 0. Reject u = −3 and retain u = 2, so x = ln 2 only. In a supplied amount model P(t) = 4e^(0.3t), the condition P = 10 becomes e^(0.3t) = 10/4, a positive dimensionless ratio. Taking ln gives t = ln(5/2)/0.3; check any stated time domain separately.

Pause and explain

For u = eˣ, the quadratic candidates are u = −3 and 2. Which can give a real x?

Put the idea to work

Worked example

Solve 3^(x + 1) = 4^(2x − 1), giving an exact answer and a three-significant-figure value.

Show the worked solution

Take ln and apply the power law: (x + 1)ln 3 = (2x − 1)ln 4.
Collect x terms to obtain x(2 ln 4 − ln 3) = ln 12.
Divide by the non-zero coefficient to get x = ln 12/(2 ln 4 − ln 3) ≈ 1.48; substitution makes the two sides equal.

Answer x = ln 12/(2 ln 4 − ln 3) ≈ 1.48.

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From guided practice to a new situation

Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.

1 · Guided

Solve 5^(2x − 1) = 7 exactly and to three significant figures.

Give me a hint

Take ln of both positive sides and solve the resulting linear equation.

Compare my reasoning

The power law gives (2x − 1)ln 5 = ln 7.
Hence 2x − 1 = (ln 7)/(ln 5) and x = [1 + (ln 7)/(ln 5)]/2.
The numerical value is approximately 1.10 to three significant figures; use the unrounded value to verify the original equation.

x = [1 + (ln 7)/(ln 5)]/2 ≈ 1.10.

Look for these in your work

Retain the complete exponent when taking logs.
Solve exactly before rounding the final value.

2 · Independent

Solve e^(2x) + eˣ − 6 = 0 over the reals.

Give me a hint

Set u = eˣ, explicitly keeping u > 0.

Compare my reasoning

The equation becomes u² + u − 6 = (u + 3)(u − 2) = 0.
Reject u = −3 because eˣ > 0; retain u = 2.
Therefore x = ln 2; the original equation gives 4 + 2 − 6 = 0.

x = ln 2 only; u = −3 is invalid.

Look for these in your work

Retain the substitution's positive range.
Check the recovered exponent in the original equation.

3 · Transfer

A supplied amount model is P(t) = 4e^(0.3t) units, with t in hours and t ≥ 0. Find when the amount reaches 10 units, exactly and to three significant figures.

Give me a hint

Divide by the initial amount to form a dimensionless exponential equation.

Compare my reasoning

Set 4e^(0.3t) = 10, giving e^(0.3t) = 5/2.
Taking ln gives 0.3t = ln(5/2), so t = ln(5/2)/0.3.
This positive time is approximately 3.05 hours, belongs to the model domain and gives P = 10 when substituted.

t = ln(5/2)/0.3 ≈ 3.05 hours.

Look for these in your work

Use the positive amount ratio before taking ln.
Check the non-negative time domain and the original amount.

Common mistakes

Taking ln moves an exponent by the power law; it does not remove the base's logarithm.
Reject non-positive candidates for a positive exponential substitution.
Keep all valid roots and check the original equation and any model domain.

Recall without your notes

What can you explain now?

Solve 2ˣ = 5 exactly.

Compare with the explanation

x = (ln 5)/(ln 2).

Taking ln gives x ln 2 = ln 5; divide by the positive non-zero value ln 2.

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Make it stick

Solve an unknown-exponent equation by the log power law, then explain why an exponential substitution imposes a positive candidate range.

If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.

Keep learning Mathematics

Check the official syllabus 2026–2027

Course scope and assessment

312 sequenced lessons provide 936 guided, independent and application problems covering the 160 objectives in sections 1.1–1.8, 2.1–2.6, 3.1–3.9, 4.1–4.5, 5.1–5.5 and 6.1–6.5. The map lists all 38 sections across six alternative components; all eight Pure Mathematics 1 sections and all six Pure Mathematics 2 sections, from Paper 2 Algebra through Numerical solution of equations, currently have full teaching sequences. Paper 3 Algebra, section 3.1, links the six shared foundation lessons plus six lessons on partial fractions and rational binomial expansions; Paper 3 logarithms, trigonometry and numerical equations use the shared lessons, with the Paper 1 graph foundation linked into trigonometry. Paper 3 differentiation combines the shared methods with its inverse-tangent derivative lesson. Paper 3 Integration links four shared standard and identity lessons plus seven advanced lessons, covering all six section 3.5 requirements. Eight Vectors lessons cover all six section 3.7 requirements, with 2D and 3D line and solid applications. Nine Differential equations lessons cover all four section 3.8 requirements, including Paper 3 integration techniques, initial conditions, equilibrium solutions and conditional model interpretations. Eleven Complex numbers lessons cover all eight section 3.9 requirements, with seven Argand diagrams and explicit argument and boundary checks. All nine Pure Mathematics 3 sections now have teaching linked to every mapped requirement. Nine Forces and equilibrium lessons cover all seven section 4.1 requirements, with six force diagrams, signed incline-friction intervals, reaction-dependent pull and push limits and separate-body third-law checks. Nine Kinematics lessons cover the four official section 4.2 requirements, with the graph requirement mapped to three relationships. Six motion diagrams support directed quantities, graph slopes and signed areas, Paper 1 time-calculus methods, reversal distances, constant-acceleration stages and shared-clock particle meetings. Six Momentum lessons cover both section 4.3 requirements, including signed system totals, conservation assumptions, outgoing velocities, coalescence, unknown masses and successive impacts. Four diagrams show system momentum through the stipulated events, and supplied kinetic-energy checks distinguish conservation of momentum from changes in kinetic energy. Twelve Newton’s laws lessons cover all four section 4.4 requirements, with vertical support and release stages, rough horizontal and incline motion, inferred parameters, hanging and surface pulley systems, rigid tow-bar thrust and slack-string transitions. Seven force diagrams support the separate-particle balances. Nine Energy, work and power lessons cover all five section 4.5 requirements, with angle-dependent work, kinetic and reference potential energy, signed net-work balances, smooth curved slides, rough and driven slopes, ideal connected-system energy and instantaneous power on hills. Five diagrams distinguish the energy inputs, losses and valid positive-speed force model. All five Mechanics sections now have teaching and staged practice linked to their mapped requirements. Thirteen Representation of data lessons cover all five section 5.1 requirements, including choice and limitations of displays, ordered and back-to-back stems, raw summaries and box plots, unequal-width density histograms, cumulative-frequency interpolation, descriptive moment calculations, grouped estimates, coding, combining two datasets and correcting totals. Six accessible diagrams retain repeated observations and distinguish numerical spread, frequency area and estimated readings. Ten Permutations and combinations lessons cover both official section 5.2 requirements, with selection and ordered roles, repeated objects, adjacency blocks, non-adjacency gaps, leading-zero cases, labelled multi-row seating and mixed restrictions. Six accessible diagrams show outcome trees, ordered versus unordered choices, repeated copies, blocks, gaps and distinct row seats. Eleven Probability lessons cover all four section 5.3 requirements, with equiprobable elementary outcomes, selection-based probabilities, complements and disjoint regions, independence tests, restricted sample spaces, two-way tables, replacement and removal trees, reversed conditions and unknown parameters. Seven accessible diagrams show dice outcomes, random subsets, event regions, a conditioned grid, a count table and two probability trees. Twelve Discrete random variables lessons cover all three official section 5.4 requirements, with complete probability tables, moment and parameter checks, binomial assumptions, exact probabilities and moments, geometric first-success probabilities, tails, expectation and integer reliability thresholds. Six accessible diagrams distinguish outcome grouping, weighted moments, fixed trials, discrete probability bars, waiting conventions and integer tail deadlines. Twelve Normal distribution lessons cover all three official section 5.5 requirements, with continuous-area interpretation, MF19 cumulative tables, full standardisation, tail and interval regions, conditional probabilities, percentiles, one- and two-parameter inference, binomial suitability, continuity corrections and approximate integer thresholds. Seven accessible diagrams show model areas, table reading, aligned scales, intervals, quantiles, matched binomial and normal shapes and whole integer cells. All five Statistics 1 sections now have teaching and staged practice linked to their mapped requirements. Fourteen Poisson distribution lessons cover all five official section 6.1 requirements, with homogeneous event-count assumptions, exposure scaling, exact masses and tails, conditional count events, moments, parameter inference, probability duration targets, disjoint and overlapping windows, rare-count binomial approximations, matched normal approximations and continuity corrections. Nine accessible diagrams show event exposures, unit conversions, discrete bars, complete tail complements, population moments, shared windows, approximate count models and integer cells. Exact checks distinguish approximate and exact close-target decisions. Fourteen Linear combinations lessons cover the six listed results in official section 6.2, with affine moments, signed scales and constant cases, expectation without independence, independent squared variance contributions, fixed and random fees, independent copies versus reused values, affine normal events, independent normal sums and differences, weighted and conditional areas, coefficient inference, exact independent Poisson sums and overlapping exposure limits. Nine accessible diagrams distinguish scale changes, variance contributions, component costs, copy mechanisms, aligned normal scales, summed areas, Poisson count pairs and unsupported distribution claims. Fourteen Continuous random variables lessons cover both official section 6.3 requirements, with density validity and units, support-aware intervals, complements and conditional regions, first and second moments, percentiles by direct density areas, symmetry checks, parameter inference, infinite-support power and exponential integrals, divergent-moment limits and combined applications. Nine accessible diagrams distinguish density height from probability, normalisation, negative-density failures, central and conditional areas, mean versus median, percentile locations and partial infinite tails. Explicit cumulative distribution function knowledge is not required. Eighteen Sampling and estimation lessons cover all eight official section 6.4 requirements, with population and frame definitions, random-number selection, sampling-quality limits, independent sample-mean moments, normal means, CLT conditions, SE scaling, raw, summarised and coded unbiased estimates, known-variance and large-sample mean intervals, repeated-sampling interpretation, precision planning, proportion intervals, sparse-outcome limits and contextual applications. Ten accessible diagrams distinguish selection frames, number rejection, repeated sample means, population versus mean density, CLT models, variance divisors, interval targets, illustrative coverage, integer size checks and proportion margins. Eighteen Hypothesis tests lessons cover all five official section 6.5 requirements, with pre-specified alternatives, inclusive count tails, explicit equal-tail conventions, integer critical regions and actual significance, exposure-aware Poisson tests, continuity-corrected count approximations, normal and large-sample mean tests, confidence-interval connections, error classifications, fixed-region binomial, Poisson and normal error probabilities and sample-size power comparisons. Nine accessible diagrams show the testing procedure, exact count regions, unequal Poisson tails, discrete significance jumps, corrected count boundaries, matching mean intervals and alternative error distributions. All shared lessons are counted once in the course total. All 38 sections now link teaching and practice. Eight earlier overview lessons remain available, but do not count as full section coverage. This is a content-availability measure, not full programme completion or exam readiness. Written practice is self-checked, not officially marked. Check the 2028–2030 syllabus separately if sitting an examination from 2028; this teaching map uses 2026–2027 requirements.