A-Level Maths AQA 7357

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 $e^{kx}$ 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:

$\log_a{x} + \log_a{y} ≡ \log_a{xy}$; $\log_a{x} - \log_a{y} ≡ \log_a{\frac{x}{y}}$; $k\log_a{x} ≡ \log_a{x^k}$

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

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.