Study

4 pages

Chapter 5: Exponentials and Logarithms

Reference: Pearson Edexcel International A Level Pure Mathematics 3 Student Book. This chapter connects exponential growth, natural logarithms and modelling with real data.

What you'll learn

Main skills from this lesson

Use e^x and ln x fluently

Apply log laws to solve equations

Linearise exponential models from data

Page 1

Exponential Functions

An exponential function has the variable in the power. The special base e appears throughout calculus.

For y = e^(ax + b) + c:

a affects growth or decay speed,

b shifts the input,

c shifts the graph vertically.

e^x is always positive

Page 2

Natural Logarithms

ln x means log base e. It reverses e^x.

ln(e^x) = x

e^(ln x) = x, for x > 0

Use ln when the unknown is in an exponent.

Remember

You cannot take ln of zero or a negative number in this course.

Page 3

Log Laws

The log laws help simplify and solve:

log(ab) = log a + log b

log(a/b) = log a - log b

log(a^n) = n log a

These work only when the log arguments are positive.

Practice

Solve e^(2x)=7 in exact form.

Take ln of both sides: 2x = ln 7, so x = (ln 7)/2.

Page 4

Exponential Modelling

For data that grows or decays by a percentage, an exponential model may fit.

A common model is y = Ae^(kx). Taking logs gives ln y = ln A + kx, which is a straight-line form.

Study tip: identify what the gradient and intercept mean in context.

ln y = kx + ln A

Great work

Continue with quizzes, flashcards, or games when you are ready.

Pure Mathematics 3 chapters