Exponentials and Logarithms worksheets with answers

Sketches exponential graphs y = aˣ for a > 1 and 0 < a < 1, with their intercept (0, 1) and asymptote y = 0, and their transformations. Students also find k and a in y = kaˣ from two points on the graph.

AQA 7357

F1, B1, B9

Edexcel 9MA0

6.1, 2.1, 2.9

Introduces e ≈ 2.718 as the base for which the gradient of y = eˣ equals y. Students learn to differentiate eᵏˣ and its multiples, and sketch transformations of y = eˣ.

AQA 7357

F1, F2, B9

Edexcel 9MA0

6.1, 6.2, 2.9

Uses models of the form P = Aeᵏᵗ to make predictions, and to interpret the initial value, the constant k and the long-term value. Students also find and interpret rates of change, and criticise and improve a model.

AQA 7357

F2, F7

Edexcel 9MA0

6.2, 6.7

Introduces logarithms: logₐ n = x means aˣ = n. Students switch between exponential and log form, evaluate logs without a calculator, solve simple log equations and use the calculator's log keys.

AQA 7357

F3, B1

Edexcel 9MA0

6.3, 2.1

Derives the three laws of logarithms from the laws of indices. Students combine logs into a single logarithm, expand logs, and solve equations using the laws, rejecting solutions that are not valid.

AQA 7357

F3, F4

Edexcel 9MA0

6.3, 6.4

Solves equations such as aˣ = b by taking logs, and uses the change of base formula. Students also solve equations with x in two powers, and quadratics in aˣ, rejecting impossible roots.

AQA 7357

F4, F5

Edexcel 9MA0

6.4, 6.5

Introduces ln x as the inverse of eˣ. Students solve equations such as eᵃˣ⁺ᵇ = p and ln(ax + b) = q exactly, solve quadratics in eˣ and in ln x, and use ln to find the constants in exponential models.

AQA 7357

F3, F4, F7

Edexcel 9MA0

6.3, 6.4, 6.7

Shows how logarithms turn y = axⁿ and y = kbˣ into straight-line graphs. Students find the constants from the gradient and intercept, and estimate them from real data.

AQA 7357

F4, F6, F7

Edexcel 9MA0

6.4, 6.6, 6.7