A-Level Maths OCR B (MEI) H640

Know and use the function $y = a^x$ and its graph.

For $a > 0$.

Know and be able to use the function $y = \ln{x}$ and its graph. Know the relationship between $\ln{x}$ and $e^x$.

*$\ln{x}$ is the inverse function of $e^x$.

Notation: $\log_e{x} = \ln{x}$ *

Be able to solve problems involving exponential growth and decay; be able to consider limitations and refinements of exponential growth and decay models.

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. Finding long term values.

Be able to convert from an index to a logarithmic form and vice versa.

$x = a^y \iff y = \log_a{x}$ for $a > 0$ and $x > 0$.

Understand a logarithm as the inverse of the appropriate exponential function and be able to sketch the graphs of exponential and logarithmic functions.

$y = \log_a{x} \iff a^y = x$ for $a > 0$ and $x > 0$. Includes finding and interpreting asymptotes.

Understand the laws of logarithms and be able to apply them, including to taking logarithms of both sides of an equation.

*$\log_a{(xy)} = \log_a{x} + \log_a{y}$$\log_a{\Big(\dfrac{x}{y}\Big) = \log_a{x} - \log_a{y}}$$\log_a{x^k} = k\log_a{x}$ Including, for example, $k = -1$ and $k = -\frac{1}{2}$.

Excluded: Change of base of logarithms.*

Know and use the values of $\log_a{a}$ and $\log_a{1}$.

$\log_a{a} = 1$, $\log_a{1} = 0$

Be able to solve an equation of the form $a^x = b$.

Includes solving related inequalities.

Know how to reduce the equations $y = ax^n$ and $y = ab^x$ to linear form and, using experimental data, to use a graph to estimate values of the parameters.

By taking logarithms of both sides and comparing with the equation $y = mx + c$. Learners may be given graphs and asked to select an appropriate model.

Know and be able to use the function $y = e^x$ and its graph.

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