Cambridge Additional Mathematics

130 Logarithms (Chapter 5)

Opening problem

In a plentiful springtime, a population of 1000 mice will double

every week.

The population aftertweeks is given by the exponential

function P(t) = 1000£ 2 t mice.

Things to think about:

a What does the graph of the population over time look

like?

b How long will it take for the population to reach20 000

mice?

c Can we write a function fortin terms ofP, which

determines the time at which the populationPis reached?

d What does the graph of this function look like?

Consider the exponential function f:x 7! 10 x

or f(x)=10x.

The graph of y=f(x) is shown alongside, along with

its inverse function f¡^1.

Sincefis defined by y=10x,

f¡^1 is defined by x=10y.

finterchangingxandyg

yis the exponent to which the base 10 is raised in order to

getx.

We write this as y= log 10 x or lgx and say thatyis thelogarithm in base 10 ,ofx.

Logarithms are thus defined to be the inverse of exponential functions:

If f(x)=10x then f¡^1 (x) = log 10 x or lgx.

WORKING WITH LOGARITHMS

For example: 10 000 = 10^4

1000 = 10^3

100 = 10^2

10 = 10^1

1=10^0

0 :1=10¡^1

0 :01 = 10¡^2

0 :001 = 10¡^3

Many positive numbers can be

easily written in the form 10 x.

A LOGARITHMS IN BASE 10

y

1 x

1

f-1

y=x

y = f(x)

O

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_05\130CamAdd_05.cdr Friday, 20 December 2013 1:04:24 PM BRIAN