#8.2
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 \(x > 0\).
#8.3
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.
#8.4
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.]
#8.5
Know and use the values of \(\log_a{a}\) and \(\log_a{1}\).
\(\log_a{a} = 1\), \(\log_a{1} = 0\)
#8.6
Be able to solve an equation of the form \(a^x = b\).
Includes solving related inequalities.
#8.7
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.
#8.9
Know that the gradient of \(e^{kx}\) is \(ke^{kx}\) and hence understand why the exponential model is suitable in many applications.
#8.10
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:
\(\log_e{x} = \ln{x}\)
#8.11
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.