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

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

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.

Gradient of exponential functions

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

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

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

Logarithms

Understand and use the laws of logarithms:

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

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

Laws of logarithms

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

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 \(y\).

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.

Exponential modelling