Know and use the function \(a^x\) and its graph, where \(a\) is positive.

Know and use the function \(e^x\) and its graph.

Examples may include the comparison of two population models or models in a biological or financial context. The link with geometric sequences may also be made.

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.

See 1.07j for explicit differentiation of \(e^x\).

Gradient of exponential functions

Know and use the definition of \(\log_a{x}\) (for \(x > 0\)) as the inverse of \(a^x\) (for all \(x\)), where \(a\) is positive.

Learners should be able to convert from index to logarithmic form and vice versa as \(a=b^c \iff c=\log_b{a} \).

The values \(\log_a{a} = 1\) and \(\log_a{1} = 0\) should be known.

Logarithms

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

Logarithms

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

e.g. In solving equations involving logarithms or exponentials.

The values \(\ln{e} = 1\) and \(\ln{1} = 0\) should be known.

Solving exponential and logarithmic equations

Understand and be able to use the laws of logarithms:

1. \(\log_a{x} + \log_a{y} = \log_a{(xy)} \)


2. \(\log_a{x} - \log_a{y} = \log_a{\Big(\dfrac{x}{y}\Big)} \)


3. \(k\log_a{x} = \log_a{x^k} \)

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

Learners should be able to use these laws in solving equations and simplifying expressions involving logarithms.

[Change of base is excluded.]

Laws of logarithms

Be able to solve equations of the form \(a^x = b\) for \(a > 0\).

Includes solving equations which can be reduced to this form such as \(2^x = 3^{2x-1}\), either by reduction to the form \(a^x = b\) or by taking logarithms of both sides.

Solving exponential and logarithmic equations

Be able to use logarithmic graphs to estimate parameters in relationships of the form \(y = ax^n\) and \(y = kb^x\), given data for \(x\) and \(y\).

Learners should be able to reduce equations of these forms to a linear form and hence estimate values of \(a\) and \(n\), or \(k\) and \(b\) by drawing graphs using given experimental data and using appropriate calculator functions.

Estimate parameters in logarithmic graphs

Understand and be able to use exponential growth and decay and use the exponential function in modelling.

Examples may include the use of \(e\) in continuous compound interest, radioactive decay, drug concentration decay and exponential growth as a model for population growth. Includes consideration of limitations and refinements of exponential models.

Exponential modelling