A-Level Maths Edexcel 9MA0

Know and use the function $a^x$ and its graph, where $a$ is positive.

Understand the difference in shape between $a < 1$ and $a > 1$.

Know and use the function $e^x$ and its graph.

To include the graph of $y = e^{ax+b} + c$

Know that the gradient of $e^{kx}$ is equal to $ke^{kx}$ and hence understand why the exponential model is suitable in many applications.

Realise that when the rate of change is proportional to the $y$ value, an exponential model should be used.

Know and use the definition of $\log_a{x}$ as the inverse of $a^x$, where $a$ is positive and $x ≥ 0$ and $a ≠ 1$.

Know and use the function $\ln{x}$ and its graph.

Know and use $\ln{x}$ as the inverse function of $e^x$.

Solution of equations of the form $e^{ax+b} = p$ and $\ln{(ax+b)} = q$ is expected.

Understand and use the laws of logarithms:

• $\log_a{x} + \log_a{y} = \log_a{(xy)}$
• $\log_a{x} - \log_a{y} = \log_a{\Big(\dfrac{x}{y}\Big)}$
• $k\log_a{x} = \log_a{x^k}$

(including, for example, $k = -1$ and $k = \frac{1}{2}$)

Includes $\log_a{a} = 1$

Solve equations of the form $a^x=b$

Students may use the change of base formula. Questions may be of the form, e.g. $2^{3x-1} = 3$

Use logarithmic graphs to estimate parameters in relationships of the form $y = ax^n$ and $y = kb^x$, given data for $x$ and $y$.

$y = ax^n$: Plot $\log{y}$ against $\log{x}$ and obtain a straight line where the intercept is $\log{a}$ and the gradient is $n$.

$y = kb^x$: Plot $\log{y}$ against $x$ and obtain a straight line where the intercept is $\log{k}$ and the gradient is $\log{b}$.

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.

Students may be asked to find the constants used in a model.

They need to be familiar with terms such as initial, meaning when $t = 0$.

They may need to explore the behaviour for large values of $t$ or to consider whether the range of values predicted is appropriate.

Consideration of an improved model may be required.