Transcript Logarithms slides from textbook

The logarithm segment of …
STROUD
Worked examples and exercises are in the text
PROGRAMME F2
INTRODUCTION
TO
ALGEBRA
STROUD
Worked examples and exercises are in the text
Programme F2: Introduction to algebra
Algebraic expressions
Powers
Logarithms
Multiplication of algebraic expressions of a single variable
Fractions
Division of one expression by another
Factorization of algebraic expressions
STROUD
Worked examples and exercises are in the text
Programme F2: Introduction to algebra
Logarithms
Powers
Logarithms
Rules of logarithms
Base 10 and base e
Change of base
Logarithmic equations
STROUD
Worked examples and exercises are in the text
Programme F2: Introduction to algebra
Logarithms
Powers
Any real number can be written as another number written raised to a
power. For example:
9  32 and 27  33
So that:
9 27  32 33  323  35  243
Here the process of multiplication is replaced by the process of relating
numbers to powers and then adding the powers – a simpler operation.
STROUD
Worked examples and exercises are in the text
Programme F2: Introduction to algebra
Logarithms
Logarithms
If a, b and c are three real numbers where:
a  bc and b 1
The power c is called the logarithm of the number a to the base b and is
written:
c  logb a spoken as c is the log of a to the base b
STROUD
Worked examples and exercises are in the text
Programme F2: Introduction to algebra
Logarithms
Rules of logarithms
The three basic rules are:
(a) loga xy  loga x  loga y
(b) loga x  y  loga x  loga y
(c) loga xn nloga x
These lead to:
(d )
(e)
(f)
(g)
STROUD
log a a 1
log a a x  x
alog x  x
loga b1/logb a
a
Worked examples and exercises are in the text
Programme F2: Introduction to algebra
Logarithms
Base 10 and base e
On a calculator there are buttons that provide access to logarithms to two
different bases, namely 10 and the exponential number e = 2.71828 …
Logarithms to base 10 are called common logarithms and are written
without indicating the base as log
Logarithms to base e are called natural logarithms and are written as ln
STROUD
Worked examples and exercises are in the text
Programme F2: Introduction to algebra
Logarithms
Change of base
The change of base formula that relates the logarithms of a number to two
different bases is given as:
logb a  loga x  logb x
STROUD
Worked examples and exercises are in the text
Programme F2: Introduction to algebra
Logarithms
Logarithmic equations
Logarithmic expressions and equations can be manipulated using the rules
of logarithms. Example:
loga x2  3loga x  2loga 4 x
 loga x2  loga x3  loga  4 x 

2
2 3
 loga  x x2 
 16 x 
 3
 loga  x 
 16 
STROUD
Worked examples and exercises are in the text