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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 33 323 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 loga a 1 loga a x x alog x x loga b1/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 log a x2 log a x3 log a 4 x 2 2 3 log a x x2 16 x 3 log a x 16 STROUD Worked examples and exercises are in the text