Higher Maths · Relationships and Calculus
Logarithms & Exponentials | Higher Maths
Logarithms and exponentials connect a lot of real-world modelling questions in Higher Maths — population growth, radioactive decay, compound interest and pH. You will use log laws to solve exponential equations and to linearise data.
Log laws (on the formula sheet)
loga(xy) = logax + logay
loga(x/y) = logax − logay
loga(xn) = n logax
Natural log and base-e
ln x = logex
eln x = x ; ln(ex) = x
Exponential growth/decay: A(t) = A0ekt
Worked example
Worked example — Solving an exponential equation
Problem: Solve 52x = 30, giving x to 3 d.p.
- Take logs of both sides.log(52x) = log 30
- Apply the power law.2x · log 5 = log 30
- Solve.x = log 30 / (2 log 5) = 1.057 (3 d.p.)
Practice questions
Try these SQA-style questions. Tap "Show answer" to check your working.
Practice questions
- Express 3 log x − 2 log y as a single log.
Show answer
log(x3/y2) - Solve e3x = 12 to 2 d.p.
Show answer
x = ln 12 / 3 ≈ 0.83 - A radioactive substance decays by 5% each year. Write the equation A(t) and find the half-life.
Show answer
A(t) = A0(0.95)t. Half-life: 0.5 = 0.95t ⇒ t = ln 0.5 / ln 0.95 ≈ 13.5 years - Simplify log2(8) + log2(2).
Show answer
3 + 1 = 4 - Solve 2x+1 = 7 to 2 d.p.
Show answer
(x + 1)log 2 = log 7 ⇒ x + 1 = log 7 / log 2 ≈ 2.81 ⇒ x ≈ 1.81
Common mistakes
Common mistakes & how to avoid them
- Confusing ln x with log10x — the base matters.
- Trying to apply log laws when there is a sum or difference inside the log argument: log(x + y) is not log x + log y.
- Forgetting that a log is only defined for positive arguments.
Frequently asked questions
Are log laws on the formula sheet?
Yes — the three main laws (product, quotient, power) are on the SQA Higher formula sheet. Natural log identities are not, so memorise those.
When do I use ln vs log?
Use ln (natural log) when the equation involves e. Use log (base 10) when convenient with calculator-style problems. Both work — answer is the same.
How is "linearising" data tested?
A relationship of the form y = kxn linearises to log y = log k + n log x. Plotting log y against log x gives a straight line of gradient n.
Related Higher Maths topics
These topics often appear together in SQA exam questions.
Need one-to-one help with Logarithms and Exponentials?
Our Glasgow Higher Maths specialists run 1-hour one-to-one sessions and 80-minute group classes — online or at our Pollokshields Tuition Centre.
Ready to enrol?
Get matched with a Glasgow specialist
Tell us the subject and level — we’ll come back the same day with availability and a clear quote.