SAJC Chapter 3 Exp and Log Functions Teacher Copy

SAJC 2022 JC 2 H1 Mathematics Exponential and Logarithmic Functions Page 1 of 14 Chapter 3 (Pure Mathematics): Exponential and Logarithmic Functions Objectives At the end of the chapter, you should be able to: (a) understand the characteristics of the indices; (b) understand the characteristics of the exponential and logarithmic functions, ex and ln x (c) understand that exy= and lnxy= are equivalent statements; (d) understand the nature of exponential growth and decay; (e) solve simple exponential and logarithmic equations; Content 1.1 Indices 1.1.1 Laws of Indices 1.1.2 Exponential Equations 1.2 A Particular Exponential Functions 1.3 Logarithmic functions 1.3.1 Definition of a Logarithm 1.3.2 Laws of Logarithm 1.4 Solving Miscellaneous Equations References New Syllabus Additional Mathematics (8th Edition), Shinglee Publishers Pte Ltd. Introduction There are many real life applications that use logarithm, from find ing the time to reach an investment goal to modeling many natural processes, particularly in living systems. We perceive loudness of sound as the logarithm of the actual sound intensity, and d B (decibels) is a logarithmic scale. We also perceive brightness of light as the logarithm of the actual light energy, and star magnitudes are measured on a logarithmic scale. Most interesting of all could be the measurement of earthquake intensity on the Richter scale. The Richter magnitudes are based on a logarithmic scale (base 10). What this means is that for each whole number you go up on the Richter scale, the amplitude of the ground motion recorded by a seismograph goes up ten times. Using this scale, a magnitude 5 earthquake would result in ten times the level of ground shaking as a magnitude 4 earthquake. Magnitude of an earthquake is log IM S =  where I is the intensity of the earthquake (measured by the amplitude of a seismograph reading taken 100 km from the epicenter of the earthquake) and S is the intensity of a ''standard earthquake'' (whose amplitude is 1 micron =10-4 cm).

SAJC 2022 JC 2 H1 Mathematics Exponential and Logarithmic Functions Page 2 of 14 1.1 Indices Background: Let us consider this example: 23 3 3= where the base is 3 and the index is 2. Index is also called the exponent. Laws of Indices If bases a and b are positive real numbers, and indices m and n are real numbers, then 1st Law m n m na a a += Example 3 5 82 2 2= 2nd Law m mn n a a a −= Example 5 5 3 2 3 2 22 2 −== 3rd Law ( ) nm mnaa = Examples ( ) 53 1522 = and ( ) 35 1522 = 4th Law ( )nnna b ab= Example ( ) ( )55552 3 2 3 6 =  = 5th Law 0 1a = Example 031= 6th Law 1n na a − = (n is a positive integer) Example 3 3 12 2 − = 7th Law ( ) m m n mnna a a== (m and n are positive integers) Example 5 3 5