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 5322 = 8th Law 1 nnaa = Example 1 3322 =

SAJC 2022 JC 2 H1 Mathematics Exponential and Logarithmic Functions Page 3 of 14 1.1.2 Exponential Equations To solve an exponential equation where both sides can be converted to the same base, use the following Equality of Indices where we equate the unknown index to the known index n. If xnaa = , then xn= , where 1, 0 ,1.a− In general, if base 0a , then 0na  for all real values of x. Consider the equation: 3 81x = . This equation is called an exponential equation, which also involves an unknown exponent or index (in this case is known as x). Example 1 Solve the following equations: (a) 3 81x = (b) 213 2 18xx −= Solution: (a) (b) 4 3 81 33 4 x x x = = = ( ) 21 1 1 3 2 18 9 2 18 9 2 18 1 xx xx x x − − − = = = =− Example 2 Solve the equation ( ) 19 1 8 3xx+ =− Solution: ( ) ( ) ( ) ( ) ( ) ( ) 1 1 2 2 9 1 8 3 9 9 8 3 1 0 9 3 8 3 1 0 9 3 8 3 1 0 xx xx x x xx + =−  + − = + − = + − = Let 3.xy= Then 29 8 1 0yy + − = . ( )( ) 29 8 1 0 9 1 1 0 yy yy + − = − + =

SAJC 2022 JC 2 H1 Mathematics Exponential and Logarithmic Functions Page 4 of 14 2 2 9 1 0 or 1 0 1 or 19 13 or 3 1 (no solution as 3 0)9 133 3 2 x x x x yy yy x − − = + = = =− = =−  ==  =− Example 3 Solve the following simultaneous equations: 124 8 xy= and 1 9 27 3 x y+ = . Solution: 124 8 xy= and 1 9 27 3 x y+ = 2 3 122 2 xy= 2 3 1 3 3 3 x y+ = 2 3 .................(1)xy+ =− ( )21 333xy−+ = 2 1 3 2 4 ..................(2) xy xy − − = −= 2 3 .................(1) 4 2 8 .................(3) 2 (2) 55 1 xy xy x x + =− − =  = = Substitute 1x= into (1) : 1 2 3 24 2 y y y + =− =− =− The solution is 1x= and 2y=− .

SAJC 2022 JC 2 H1 Mathematics Exponential and Logarithmic Functions Page 5 of 14 Exercise 1: 1. Solve the following simultaneous equations: 139 3 xy= and 1 8 8 2 x y+ = . Solution: 139 3 xy= and 1 8 8 2 x y+ = 213 3 3xy −= 3 1 2 8 2 x y+ = 2 1 .................(1)xy+ =− ( )31 322xy−+ = 3 1 3 3 4 ..................(2) xy xy − − = −= 2 1 .................(1) 6 2 8 .................(3) 2 (2) 77 1 xy xy x x + =− − =  = = Substitute 1x= into (1) : 1 2 1 22 1 y y y + =− =− =− The solution is 1x= and 1y=− . 2. The equation of a curve is given by xy ka= , where a and k are constants. Given that the curve passes through (2, 16), (3, 32) and (5, p), find the values of a, k and p. Solution: xy ka= 2 2 At point (2,16), 2, 16 16 16 ...........................(1) x xy y ka ka k a == = = =

SAJC 2022 JC 2 H1 Mathematics Exponential and Logarithmic Functions Page 6 of 14 3 2 2 At point (3,32), 3, 32 1632 from (1) 2 ...........................(2) 16 4 2 x xy y ka a a a k == = = =  = = 5 2 At point (5, ) 5, 16 2 from (1) and (2) 2 128 x p x y p y ka p p == = = = 1.2 A Particular Exponential Function: exy= , x Background exy= is also known as the exponential function The expression ex is a function with a number of applications. The value of e is 2.718281…. It is a value arising from an attempt to perform a Calculus operation known as differentiation of the logarithmic function (see 1.3 below). 1.3 Logarithmic Functions Background: If we need to solve the equation 10 100x = we need only to rewrite the equation as 210 10x = , and arrive at the solution 2x= . What do we do if we need to solve an equation like this: 10 7x = ? To solve for x, we convert the exponential equation to a logarithmic equation. 1.3.1 Definition of a Logarithm For x > 0 and b > 0, b ≠ 1, logbyx= is equivalent to ybx = . Where b is the base and y is the index or exponent.

SAJC 2022 JC 2 H1 Mathematics Exponential and Logarithmic Functions Page 7 of 14 Note (i) Logarithms with base 10 is known as common log: 10log lgy x x== In particular, 10log 10 10 10 1 yy y =  = = (ii) Logarithms with base e is known as natural log: elog lny x x== In particular, elog e e e 1 yy y =  = = Example 4 1. Convert the following from exponential equation to logarithmic equation. (i) 34 64= (ii) 45 625= ( Verify the answer with your calculator!) 2. Solve the equation lg 2 3x= . Solution: 1i) 3 44 64 3 log 64=  = (i