Exponentials and Logarithms

Scheme of work: Year 12 A-Level: Pure 1: Exponentials and Logarithms

Prerequisite Knowledge

  • From GCSE
    • Calculate with roots, and with integer and fractional indices
    • Simplifying expressions involving sums, products and powers, including the laws of indices
    • Calculate exactly with surds
    • Simplify surd expressions involving squares and rationalise denominators
  • From AS-Level Mathematics
  • Solve quadratic equations (including those that require rearrangement) algebraically by:
    • factorising,
    • completing the square using the quadratic formula;
    • Find approximate solutions using a graph.

Success Criteria

  • Know and use the function ax and its graph, where a is positive
  • Know and use the function ex and its graph.
  • Know that the gradient of ekx equals kekx and understand why the exponential model is suitable in many applications.
  • Know and use the definition of Loga x as the inverse of ax, where a is positive and x ≥ 0.
  • Know and use the function Ln x and its graph
  • Know and use Ln x as the inverse function of ex.
  • Understand and use the laws of logarithms
\log _{a} x+\log _{a} y=\log _{a}(x y)
\log _{a} x-\log _{a} y=\log _{a}\left(\frac{x}{y}\right)
k \log _{a} x=\log _{a} x^{k}
  • Solve equations of the form ax = b.
  • Use logarithmic graphs to estimate parameters in relationships of the form y = axn and y = kbx, given data for x and y.
  • Understand and use exponential growth and decay; use in modelling (examples may include the use of e in continuous compound interest, radioactive decay, drug concentration decay, and exponential growth as a model for population growth); consideration of limitations and refinements of exponential models] 

Key Concepts

  • Logarithms are used to solve equations where the variable is a power
  • You can only take a logarithm of a positive number.
  • Logarithms in base 10 are written as log x. 
  • e is approximately 2.718… 
  • 10x and Log10x are opposite functions.
  • ex and Ln x are also opposite functions.
  • The laws of logarithms and special cases

If y = ax then x = Loga y

\log _{a} x+\log _{a} y=\log _{a}(x y)
\log _{a} x-\log _{a} y=\log _{a}\left(\frac{x}{y}\right)
k \log _{a} x=\log _{a} x^{k}
\log _{a}\left(\frac{1}{x}\right)=-\log _{a} x
e^{\ln x}=\operatorname{Ln}\left(e^{x}\right)=x
  • Equations of the form a2x can be treated as a quadratic equation.
  • If y = kbx then
\log y=x \log b+\log k

The graph of Log y against x has gradient Log b and intercept Log k.

Common Misconceptions

  • When solving simple equations of form 3x = 8 students often show correct work but lose accuracy marks with their final answer.
  • Many students struggle to transform an exponential model into the form y = mx + c.
  • Some students confuse the exponential graph with that of a parabola or cubic, failing to appreciate that y = ax has a y-intercept of 1.
  • When setting up quadratics from equations of the type e2x – 7ex + 12 = 0, students attempt to solve using the quadratic formula when factorisation is more efficient.
  • Some students misuse the laws of logarithms; for instance, when attempting to use the multiplication and division laws, they incorrectly write Log (x + 15) = Log x + Log 15 or Log (x – 2) as Log(x/2).

Exponentials and Logarithms Resources

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