A-level Mathematics₇₃₅₇

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

Know and use the function ${\mathrm{e}}^{\mathit{x}}$ and its graph.

Know that the gradient of ${\mathrm{e}}^{\mathit{k}\mathit{x}}$ is equal to $\mathit{k}{\mathrm{e}}^{\mathit{k}\mathit{x}}$ and hence understand why the exponential model is suitable in many applications.

Know and use the definition of ${\mathsf{log}}_{\mathit{a}}\mathsf{}\mathit{x}$ as the inverse of ${\mathit{a}}^{\mathit{x}}$ , where $\mathit{a}$ is positive and $\mathit{x}\mathsf{\ge }\mathsf{0}$

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

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

Understand and use the laws of logarithms:

${\mathsf{log}}_{\mathit{a}}\mathsf{}\mathit{x}\mathsf{+}{\mathsf{log}}_{\mathit{a}}\mathsf{}\mathit{y}\mathsf{\equiv }{\mathsf{log}}_{\mathit{a}}\mathsf{}\mathsf{\left(}\mathit{x}\mathit{y}\mathsf{\right)}$ ; ${\mathsf{log}}_{\mathit{a}}\mathsf{}\mathit{x}\mathsf{-}{\mathsf{log}}_{\mathit{a}}\mathsf{}\mathit{y}\mathsf{\equiv }{\mathsf{log}}_{\mathit{a}}\mathsf{}\left(\frac{\mathit{x}}{\mathit{y}}\right)$ ; $\mathit{k}{\mathsf{log}}_{\mathit{a}}\mathsf{}\mathit{x}\mathsf{\equiv }{\mathsf{log}}_{\mathit{a}}\mathsf{}{\mathit{x}}^{\mathit{k}}$

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

Solve equations of the form ${\mathit{a}}^{\mathit{x}}\mathsf{=}\mathit{b}$

Use logarithmic graphs to estimate parameters in relationships of the form $\mathit{y}\mathsf{=}\mathit{a}{\mathit{x}}^{\mathit{n}}$ and $\mathit{y}\mathsf{=}\mathit{k}{\mathit{b}}^{\mathit{x}}$ , given data for $\mathit{x}$ and $\mathit{y}$ .

Understand and use exponential growth and decay; use in modelling (examples may include the use of ${\mathrm{e}}^{\mathit{}}$ 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.