A-Level Maths Specification

OCR B (MEI) H640

Section 8: Exponentials and logarithms

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#8.1

Know and use the function \(y = a^x\) and its graph.

For \(a > 0\).

Exponential functions

#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 \(x > 0\).

Logarithms

#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.

Logarithms

#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.]

Laws of logarithms

#8.5

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

\(\log_a{a} = 1\), \(\log_a{1} = 0\)

Laws of logarithms

#8.6

Be able to solve an equation of the form \(a^x = b\).

Includes solving related inequalities.

Solving exponential and logarithmic equations

#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.

Estimate parameters in logarithmic graphs

#8.8

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

Exponential functions

#8.9

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

Gradient of exponential functions

#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:
\(\log_e{x} = \ln{x}\)

Logarithms

#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.

Exponential modelling