Logarithms, explained with examples

By The STEM-Tutor team · Updated October 2026

A logarithm answers the question: what power do I need? Logs undo exponentials, which makes them essential for solving equations, modelling growth and decay, and turning curves into straight lines.

What a logarithm is

log_a b = c means a^c = b. In words: the logarithm of b to base a is the power you raise a to in order to get b. So log₂ 8 = 3 because 2³ = 8, and log₁₀ 1000 = 3 because 10³ = 1000.

Some values to know immediately: log_a a = 1, log_a 1 = 0, and log_a (a^x) = x. You can only take logs of positive numbers, and the base a must be positive and not equal to 1.

The laws of logarithms

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

These come straight from the laws of indices: multiplying powers adds the exponents, so logs of products become sums. Note what the laws do not say: log(x + y) is not log x + log y, and log x / log y is not log(x/y).

Natural logarithms

The natural logarithm, ln x, is the log to base e, where e ≈ 2.718. It is the inverse of e^x: ln(e^x) = x and e^{ln x} = x. The function e^x is special because its gradient equals its value, which is why it appears in natural growth and decay and throughout calculus.

Worked examples

Example 1: evaluating

Find log₃ 81 and log₄ 2 without a calculator.

• 3⁴ = 81, so log₃ 81 = 4.
• 4^1/2 = 2, so log₄ 2 = 1/2.

Example 2: combining logs

Write 2 log 5 + log 4 − log 2 as a single logarithm.

• 2 log 5 = log 25. Then log 25 + log 4 − log 2 = log(25 × 4 / 2).
• = log 50.

Example 3: an exponential equation

Solve 5^x = 40, to 3 significant figures.

• Take logs: x log 5 = log 40, so x = log 40 / log 5.
• x ≈ 2.29.

Example 4: unknowns on both sides

Solve 3^{2x − 1} = 2^{x + 1}.

• Take logs: (2x − 1) ln 3 = (x + 1) ln 2.
• x(2 ln 3 − ln 2) = ln 2 + ln 3, so x = ln 6/ln 4.5.
• x ≈ 1.19.

Example 5: a hidden quadratic

Solve 4^x − 6 × 2^x + 8 = 0.

• Let u = 2^x, so 4^x = u²: u² − 6u + 8 = 0, giving u = 2 or 4.
• 2^x = 2 gives x = 1; 2^x = 4 gives x = 2. x = 1 or x = 2.

Example 6: a log equation

Solve log₂(x + 3) + log₂(x − 3) = 4.

• Combine: log₂(x² − 9) = 4, so x² − 9 = 16 and x = ±5.
• x = −5 makes the logs undefined, so x = 5. Always check solutions in the original equation.

Example 7: natural logs

Solve e^2x = 7, and solve ln(3x) = 2.

• 2x = ln 7, so x = ½ ln 7 ≈ 0.973.
• 3x = e², so x = e²/3 ≈ 2.46.

Exponential models

Many real quantities grow or decay by a constant factor over equal time intervals, such as money with compound interest, bacteria in a culture or the temperature difference of a cooling drink. These are modelled by N = N₀e^kt, with k positive for growth and negative for decay. Logs let you find the time to reach a value: for example, the time for a quantity to double is ln 2/k. Questions often finish by asking you to criticise the model, such as noting that real growth cannot continue for ever.

Graphs of logs and exponentials

y = e^x passes through (0, 1), is always positive and has the x-axis as an asymptote. y = ln x is its reflection in the line y = x: it passes through (1, 0), is defined only for x > 0, and has the y-axis as an asymptote. Every log graph y = log_a x with a > 1 has the same shape and passes through (1, 0).

Using logs to find straight lines

A favourite A level question gives experimental data and asks you to find a model. Logs turn two kinds of curve into straight lines:

• y = axⁿ: taking logs gives log y = n log x + log a. Plot log y against log x: the gradient is n and the intercept is log a.
• y = kb^x: taking logs gives log y = x log b + log k. Plot log y against x: the gradient is log b and the intercept is log k.

Example 8: finding a model

A graph of log₁₀ y against x is a straight line through (0, 0.3) with gradient 0.2. Find the model y = kb^x.

• log k = 0.3, so k = 10^0.3 ≈ 2.00. log b = 0.2, so b = 10^0.2 ≈ 1.58.
• y ≈ 2.00 × 1.58^x.

Common mistakes

• Writing log(x + y) = log x + log y.
• Forgetting that logs of zero or negative numbers are undefined, and keeping invalid solutions.
• Dropping brackets: (2x − 1) ln 3 is not 2x − 1 ln 3.
• Mixing up the two straight-line models and their axes.

Practice questions

Try each one before opening the answer.

Frequently asked questions

Are logarithms in AS or A level Maths?

Logarithms, exponentials and the straight-line models are in the first year of A level Maths for all the main boards.

Are logarithms in AP Calculus?

AP Calculus uses ln x and e^x constantly, especially their derivatives and integrals. Precalculus covers the log laws in depth.

Key takeaways

• log_a b = c means a^c = b.
• Product, quotient and power laws come from the laws of indices.
• ln is the inverse of e^x.
• Take logs of both sides to solve for an unknown power.
• Logs turn y = axⁿ and y = kb^x into straight lines.

Related guides

• Differential equations
• A level Maths topics
• A level Maths formula booklet