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**A-Level Mathematics Year **1: Exponentials and Logarithms

Throughout the unit, students progress from understanding how exponents can be written using logarithms to transforming exponential models into straight-line graphs.

**Lessons on Integration**

**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 a
^{x}and its graph, where a is positive - Know and use the function e
^{x}and its graph. - Know that the gradient of ek
^{x}is equal to ke^{kx }and hence understand why the exponential model is suitable in many applications. - Know and use the definition of Log
_{a}x as the inverse of a^{x}, 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 e
^{x}. - 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 a
^{x}= b. - Use logarithmic graphs to estimate parameters in relationships of the form y = ax
^{n}and y = kb^{x}, 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, exponential growth as a model for population growth); consideration of limitations and refinements of exponential models]

**Teaching Points**

*Logarithms answer the question: ‘To what power do I need to raise to get ?’**You can only take a logarithm of a positive number.**Logarithms in base 10 are written as log x.*- e is approximately 2.718…

*10*^{x}and Log10^{x}are opposite functions.

*e*^{x}and Ln x are also opposite functions.

- The laws of logarithms and special cases

If y = a^{x} then x = Log_{a} 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 a
^{2x}can be treated as a quadratic equation.

- If y = kb
^{x}then

\log y=x \log b+\log k

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

**Misconceptions**

- When solving simple equations of the form 3
^{x}= 8 students often show correct working 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 = a
^{x}has y-intercept of 1. - When setting up quadratics from equations of the type e
^{2x}– 7e^{x}+ 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).

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