Home → Curriculum Hub → Maths Lessons → Schemes of Work → A-Level Mathematics → Exponentials and Logs
This scheme of work for A-Level exponentials and logarithms builds on earlier work at GCSE with index laws and simple growth by extending these skills to the exponential function, natural logarithms and equations where the unknown is a power. It develops the skill of moving between exponential and logarithmic form, and these ideas are used throughout the AS Pure course.
Students learn to sketch y equals a to the x, then the graph of y equals e to the x and its gradient. They interpret growth and decay models, then use logarithms as the inverse of powers. They simplify with the laws of logarithms, then solve equations that involve logs or exponentials.
By the end of this unit students will be able to:
Students should be secure with the following before beginning this unit.
For y equals a to the x, with a positive and not 1, every y-value is positive and the x-axis is a horizontal asymptote. If a is greater than 1 the graph increases. If a is between 0 and 1 it decreases. It is not a parabola or a cubic.
e is about 2.718. The graph of y equals e to the x passes through (0, 1) and its gradient at each point equals its y-value. For y equals e to the kx the gradient is k times e to the kx.
If y equals a to the x then x equals log base a of y. You can only take a logarithm of a positive number. log x means base 10. ln x means base e. 10 to the x and log x undo each other. e to the x and ln x undo each other.
The log of a product is the sum of the logs. The log of a quotient is the difference of the logs. A multiple in front of a log becomes a power inside. Equations such as e to the 2x minus 7 e to the x plus 12 equals 0 become quadratics after the substitution u equals e to the x. Check every solution makes each log argument positive.
| Misconception | Teaching focus |
|---|---|
| Sketching y equals a to the x as a parabola or a cubic, with the wrong intercept. | The graph crosses the y-axis at 1, stays positive, and does not turn back. |
| Rounding too early when solving an equation such as 3 to the x equals 8. | Take logs of both sides and keep log 8 over log 3 until the final rounding. |
| Using the quadratic formula on e to the 2x minus 7 e to the x plus 12 equals 0 before a substitution. | Let u equal e to the x. The equation becomes a quadratic in u that factorises. |
| Applying a logarithm law to a sum that is not a product. | Combine logs only with the product, quotient and power laws, then rewrite as a power. |
| Accepting a solution that makes a logarithm argument zero or negative. | Check every answer in the original equation. Logarithms are only defined for positive arguments. |
Every lesson in this unit is already built. A Mr Mathematics membership gives you the presentation, worksheet, question generator and video tutorial for all six lessons, alongside the full Key Stage 3, GCSE, IGCSE and A-Level libraries.
A school membership covers every teacher in your department, so the whole scheme of work is resourced from one subscription.
Students learn how to use function notation and transformations to sketch exponential graphs.
Later, as learning progresses, students reverse these transformations to determine the equation of a transformed equation
Students learn how to sketch the properties of an exponential functions of the form y = ex.
Later, as learning progresses they differentiate equations of the form y = A + bxt.
Students learn how to use and interpret models that use exponential functions.
Later, as learning progresses solve differentiation equations to model real-life problems.
Students learn about the relationship between exponents and logarithms.
Later, as learning progresses they set up and solve equations involving logarithms.
Students learn how to simplify exponents using the laws of logarithms.
Later, as learning progresses they use multiplication, division and power laws to solve equations.
Students learn how to solve equations using the laws of logarithms.
Later, as learning progresses they solve log equations disguised as quadratics.
Solving Equations with Logs - Part 1
Solving Equations with Logs - Part 2
Solving Equations with Logs - Part 3
Students learn how to solve equations involving e^x and Ln (x).
Later, as learning progresses they solve equations involving e^x and Ln (x) as disguised quadratics.
Students learn how to transform exponential models into a straight-line graph.
Later, as learning progresses they use logarithms to find the half-life or radioactive materials.
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