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 \(y = e^{ax+b} + c\)

Exponential functions

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.

Gradient of exponential functions

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.

Logarithms

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 \(\log_a{a} = 1\)

Laws of logarithms

Solve equations of the form \(a^x=b\)

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

Solving exponential and logarithmic equations

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 \(\log{b}\).

Estimate parameters in logarithmic graphs

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 a second improved model may be required.

Exponential modelling