Exponential and Logarithmic Functions 5 Exponential and Logarithmic Functions Case Study 5.1 Rational Indices 5.2 Logarithmic Functions 5.3 Using Logarithms to Solve Equations 5.4 Graphs of Exponential and Logarithmic Functions 5.5 Applications of Logarithms Chapter Summary

Case Study The growth rate of bacteria can be expressed by an exponential function. We can use it to find the number of bacteria. How can we find the number of bacteria after a certain period of time? Although it is not easy to see bacteria with our naked eyes, bacteria do exist almost everywhere. Most bacteria reproduce by cell division, that is, one bacterium can divide and become 2, 2 become 4, and so on. However, each species reproduces itself at different rates due to different humidities and temperatures.

5.1 Rational Indices A. Radicals We learnt that if x2  y, for y  0, then x is a square root of y. If x3  y, then x is a cube root of y. If x4  y, then x is a fourth root of y. is usually written as . In general, for a positive integer n, if xn  y, then x is an nth root of y and we use the radical to denote an nth root of y. Remarks:  If n is an even number and y  0, then y has one positive nth root and one negative nth root. Positive nth root: ; Negative nth root:  If n is an even number and y  0, Examples: then is not a real number.  If n is an odd number, then  0 (for  0) and  0 (for y  0).

5.1 Rational Indices B. Rational Indices In junior forms, we learnt the laws of indices for integral indices. Recall that (am)n  amn, where m and n are integers. Try to find the meaning of and : Since  y Take nth root on both sides, we have . Then consider and Hence we have .

5.1 Rational Indices B. Rational Indices Therefore, for y  0, we define rational indices as follows: 1. 2. where m, n are integers and n  0. Remarks:  In the above definition, y is required to be positive. However, the definition is still valid for y  0 under the following situation: The fraction is in its simplest form and n is odd. For example:

Example 5.1T 5.1 Rational Indices Solution: B. Rational Indices Simplify , where b  0 and express the answer with a positive index. Solution: First express the radical with a rational index. Then use (bm)n  bmn to simplify the expression.

Example 5.2T 5.1 Rational Indices Solution: B. Rational Indices Evaluate . Solution: Change the mixed number into an improper fraction first.

Example 5.3T 5.1 Rational Indices Solution: B. Rational Indices Simplify . Solution: Change the numbers to the same base before applying the laws of indices.

5.1 Rational Indices C. Using a Calculator to Find  y m n The following is the key-in sequence of finding : 5 32 Since , we can also press the following keys in sequence: 32 2 5 Remarks: If n is an even number and ym  0, then ym has two nth real roots, i.e., . However, the calculator will only display .

5.1 Rational Indices D. Using the Law of Indices to Solve Equations For the equation  b, where b is a non-zero constant, p and q are integers with q  0, we can take the power of on both sides and solve the equation:

Example 5.4T 5.1 Rational Indices Solution: D. Using the Law of Indices to Solve Equations Example 5.4T If , find x. Solution: Change the equation into the form first.

Example 5.5T 5.1 Rational Indices Solution: D. Using the Law of Indices to Solve Equations Example 5.5T Solve 22x  1  22x  8. Solution: 22x  1  22x  8 2(22x)  22x  8 Take out the common factor 22x first. 22x(2  1)  8 22x  8 22x  23 2x  3 x 

5.2 Logarithmic Functions A. Introduction to Common Logarithm If a number y can be expressed in the form ax, where a  0 and a  1, then x is called the logarithm of the number y to the base a. It is denoted by x  loga y. If y  ax, then loga y  x, where a  0 and a  1. Notes:  If y  0, then loga y is undefined. Thus the domain of the logarithmic function loga y is the set of all positive real numbers of y.  When a  10 (base 10), we write log y for log10 y. This is called the common logarithm. In the calculator, the button log also stands for the common logarithm. If y  10n, then log y  n. ∴ log 10n  n for any real number n.

5.2 Logarithmic Functions A. Introduction to Common Logarithm By the definition of logarithm and the laws of indices, we can obtain the following results directly:  ∵  102 ∴ log  2  ∵  101 ∴ log  1  ∵ 1  100 ∴ log 1  0  ∵ 10  101 ∴ log 10  1  ∵ 100  102 ∴ log 100  2  ∵ 1000  103 ∴ log 1000  3 Values other than powers of 10 can be found by using a calculator. For example:  log 34  1.5315 (cor. to 4 d. p.)  Given log x  1.2. ∴ x  101.2  0.0631 (cor. to 3 sig. fig.)

5.2 Logarithmic Functions B. Basic Properties of Common Logarithm The function f (x)  log x, for x  0 is called a logarithmic function. There are 3 important properties of logarithmic functions: In general, 1. log M  log N  log (MN); 2. ; 3. (log M)n  log M n. For M, N  0, 1. log (MN)  log M  log N 2. log  log M  log N 3. log M n  n log M Let M  10a and N  10b. Then log M  a and log N  b.  Consider MN  10a  10b  10a  b ∴ log (MN)  a  b  log M  log N Take common logarithm on both sides.  Consider   10a  b  Consider M n  (10a)n  10na ∴ log  a  b  log M  log N ∴ log M n  na  n log M

Example 5.6T 5.2 Logarithmic Functions Solution: B. Basic Properties of Common Logarithm Example 5.6T Evaluate the following expressions. (a) log 5  log (b) Solution: (a) Since 3 log 5  log 80  log 53  log 80  log (53  80)  log 10 000  4 ∴

Example 5.6T 5.2 Logarithmic Functions Solution: B. Basic Properties of Common Logarithm Example 5.6T Evaluate the following expressions. (a) log 5  log (b) Solution: (b)

Example 5.7T 5.2 Logarithmic Functions Solution: B. Basic Properties of Common Logarithm Example 5.7T Simplify , where x  0. Solution:

Example 5.8T 5.2 Logarithmic Functions Solution: B. Basic Properties of Common Logarithm Example 5.8T If log 3  a and log 5  b, express the following in terms of a and b. (a) log 225 (b) log 18 Solution: (a) log 225  log (32  52) (b) log 18  log (2  32)  log 32  log 52  log 2  log 32  2 log 3  2 log 5  log  2 log 3  2a  2b  log 10  log 5  2 log 3  1  b  2a

5.2 Logarithmic Functions C. Other Types of Logarithmic Functions We have learnt the logarithmic function with base 10 (i.e. common logarithm). For logarithmic functions with bases other than 10, such as the function f (x)  loga x for x  0, a  0 and a  1, they still have the following properties: For M, N, a  0 and a  1, 1. loga a  1 2. loga 1  0 3. loga (MN)  loga M  loga N 4. loga  loga M  loga N 5. loga M n  n loga M

Example 5.9T 5.2 Logarithmic Functions Solution: C. Other Types of Logarithmic Functions Example 5.9T Evaluate . Solution:

5.2 Logarithmic Functions D. Change of Base Formula A calculator can only be used to find the values of common logarithm. For logarithmic functions with bases other than 10, we need to use the change of base formula to transform the original logarithm into common logarithm: Change of Base Formula For any positive numbers a and M with a  1, we have . Let y  loga M, then we have a y  M. log a y  log M Take common logarithm on both sides. y log a  log M

5.2 Logarithmic Functions D. Change of Base Formula In fact, besides common logarithm, the change of base formula can also be applied for logarithms with bases other than 10. Change of Base Formula For any positive numbers a, b and M with a, b  1, we have .

Example 5.10T 5.2 Logarithmic Functions Solution: D. Change of Base Formula Example 5.10T Solve the equation logx  1 8  25. (Give the answer correct to 3 significant figures.) Solution: logx  1 8  25  25 log (x  1)   0.03612 x  1  1.08673 x  0.0867 (cor. to 3 sig. fig.)

Example 5.11T 5.2 Logarithmic Functions Solution: D. Change of Base Formula Example 5.11T Show that log8 x  log4 x, where x  0. Solution: log8 x

5.3 Using Logarithms to Solve Equations A. Logarithmic Equations Logarithmic equations are equations containing the logarithm of one or more variables. For example:  log x  2  log5 (x  2)  1 We need to use the definition and the properties of logarithm to solve logarithmic equations. For example: If loga x  2, then x  a2 .

Example 5.12T 5.3 Using Logarithms to Solve Equations Solution: A. Logarithmic Equations Example 5.12T Solve the equation log3 (x  7)  log3 (x  1)  2. Solution: log3 (x  7)  log3 (x  1)  2 x  7  9x  9 x  2

Example 5.13T 5.3 Using Logarithms to Solve Equations Solution: A. Logarithmic Equations Example 5.13T Solve the equation log x2  log 4x  log (x  1). Solution: log x2  log 4x  log (x  1)  log 4x(x  1)  log (4x2  4x) x2  4x2  4x 3x2  4x  0 x(3x  4)  0 x  or 0 (rejected) When x  0, log x2  log 0, log 4x  log 0 and log (x  1)  log (1), which are undefined. So we have to reject the solution of x  0.

5.3 Using Logarithms to Solve Equations B. Exponential Equations Exponential equations are equations in the form ax  b, where a and b are non-zero constants and a  1. To solve ax  b: ax  b log ax  log b Take common logarithm on both sides. x log a  log b The equation is reduced to linear form. ∴ x 

Example 5.14T 5.3 Using Logarithms to Solve Equations Solution: B. Exponential Equations Example 5.14T Solve the equation 5x  3  8x  1. (Give the answer correct to 2 decimal places.) Solution: 5x  3  8x  1 log 5x  3  log 8x  1 (x  3) log 5  (x  1) log 8 x(log 5  log 8)  3 log 5  log 8 x   14.70 (cor. to 2 d. p.)

5.4 Graphs of Exponential and Logarithmic Functions A. Graphs of Exponential Functions For a  0 and a  1, a function y  ax is called an exponential function, where a is the base and x is the exponent. Consider the exponential function y  2x. y  y  2x x –1 1 2 3 4 5 y 0.5 8 16 32 The domain of the function is all real numbers. y  0 for all real values of x. Also plot the function y  : x –5 –4 –3 –2 –1 1 y 32 16 8 4 2 0.5 y-intercept  1 The graphs are reflectionally symmetric about the y-axis.

5.4 Graphs of Exponential and Logarithmic Functions A. Graphs of Exponential Functions Properties of the graphs of exponential functions: 1. The domain of exponential function is the set of all real numbers. 2. The graph does not cut the x-axis, that is, y  0 for all real values of x. 3. The y-intercept is 1. 4. The graphs of y  ax and y  are reflectionally symmetric about the y-axis. 5. For the graph of y  ax, (a) if a  1, then y increases as x increases. (b) if 0  a  1, then y decreases as x increases. The graphs of y  f (x) and y  g(x) are reflectionally symmetric about the y-axis if g(x)  f (x). Notes: Property 4 can be proved algebraically.

5.4 Graphs of Exponential and Logarithmic Functions A. Graphs of Exponential Functions Consider the graphs of y  2x and y  3x. x –1 1 2 3 4 y 0.5 8 16 y  3x y  2x x –1 1 2 3 4 y 0.33 9 27 81 The graph y  3x increases more rapidly. Consider the graphs of y  and y  . Which graph decreases more rapidly?

5.4 Graphs of Exponential and Logarithmic Functions A. Graphs of Exponential Functions Properties of the graphs of exponential functions: For the graphs of y  ax and y  bx, where a, b, x  0, (i) If a  b  1, then the graph of y  ax increases more rapidly as x increases; (ii) If 1  b  a, then the graph of y  ax decreases more rapidly as x increases.

5.4 Graphs of Exponential and Logarithmic Functions B. Graphs of Logarithmic Functions Consider the graphs of y  log2 x and y  log1 x. 2 x 0.5 1 2 4 8 16 y –1 3 x 0.5 1 2 4 8 16 y –1 –2 –3 –4 x-intercept  1 The domain of the function is all positive real numbers. The graphs are reflectionally symmetric about the x-axis.

5.4 Graphs of Exponential and Logarithmic Functions B. Graphs of Logarithmic Functions Properties of the graphs of logarithmic functions: 1. The domain of logarithmic function is the set of all positive real numbers, i.e., undefined for x  0. 2. The graph does not cut the y-axis, (that is, x  0 for all real values of y). 3. The x-intercept is 1. 4. The graphs of y  loga x and y  log1 x, where a  0 are a reflectionally symmetric about the x-axis. 5. For the graph of y  loga x, (a) if a  1, then y increases as x increases. (b) if 0  a  1, then y decreases as x increases.

5.4 Graphs of Exponential and Logarithmic Functions B. Graphs of Logarithmic Functions Consider the graph of y  log2 x. If a positive value of x is given, then the corresponding value of y can be found  by the graphical method. e.g., when x  7, y  2.8.  by the algebraic method. e.g., when x  7, y  log2 7  2.8 If a value of y is given:  Graphical method e.g., when y  1.6, x  3.0. Given a value of y, then x  2y. independent variable dependent variable  Algebraic method e.g., when y  1.6, 1.6  log2 x 21.6  x ∴ f (x)  2x f (x)  log2 x inverse function x  3.0

5.4 Graphs of Exponential and Logarithmic Functions C. Relationship between the Graphs of Exponential and Logarithmic Functions Consider the graphs of y  2x and y  log2 x. y  x Each of the functions f (x)  2x and y  log2 x is the inverse function of each other. The graphs of y  2x and y  log2 x are reflectional images of each other about the line y  x.

5.4 Graphs of Exponential and Logarithmic Functions D. Transformations on the Graphs of Exponential and Logarithmic Functions In Chapter 4, we learnt about the transformations of the graphs of functions. We can also transform logarithmic functions and exponential functions. For example: Let f (x)  log2 x and g(x)  log2 x  2. y  log2 x  2 ∴ y  f (x) is translated 2 units upwards to become y  g(x). However, we have to pay attention to the properties of logarithmic functions such as loga (MN)  loga M  loga N. For example: If h(x)  log2 4x, then it is the same as g(x): log2 4x  log2 4  log2 x  2  log2 x

Example 5.15T 5.4 Graphs of Exponential and Logarithmic Functions D. Transformations on the Graphs of Exponential and Logarithmic Functions Example 5.15T The following figure shows the graph of y  log2 x. Use the graph to sketch the graphs of the following functions: (a) y  log2 x (b) y  log2 Solution: (a) Since y  log2 x  log2 x  log2 y  log2  x 1 4  log2 x – 2 ∴ the graph of y  log2 x is obtained by translating the graph of y  log2 x two units downwards.

Example 5.15T 5.4 Graphs of Exponential and Logarithmic Functions D. Transformations on the Graphs of Exponential and Logarithmic Functions Example 5.15T The following figure shows the graph of y  log2 x. Use the graph to sketch the graphs of the following functions: (a) y  log2 x (b) y  log2 Solution: (b) Since y  log2  log2 2 – log2 x  1 – log2 x i.e., y  –log2 x  1. y  log2  2 x ∴ The graph of y  log2 is obtained by reflecting the graph of y  log2 x about the x-axis, then translating one unit upwards.

Example 5.16T 5.4 Graphs of Exponential and Logarithmic Functions D. Transformations on the Graphs of Exponential and Logarithmic Functions Example 5.16T The following figure shows the graph of y  2x. Use the graph to sketch the graphs of the following functions: (a) y  2x  2 (b) y  2x Solution: (a) Let f (x)  2x, g(x)  2x  2. y  2x + 2 ∵ g(x)  f (x  2) ∴ the graph of y  2x  2 is obtained by translating the graph of y  2x two units to the left.

Example 5.16T 5.4 Graphs of Exponential and Logarithmic Functions D. Transformations on the Graphs of Exponential and Logarithmic Functions Example 5.16T The following figure shows the graph of y  2x. Use the graph to sketch the graphs of the following functions: (a) y  2x  2 (b) y  2x Solution: y  2x (b) Let f (x)  2x, h(x)  2x. ∵ h(x)  f (–x) ∴ the graph of y  2x is obtained by reflecting the graph of y  2x about the y-axis.

5.5 Applications of Logarithms (a) Loudness of Sound Decibel (dB): unit for measuring the loudness L of sound: L  10 log where I is the intensity of sound and I0 ( 1012 W/m2) is the threshold of hearing (minimum audible sound intensity) for a normal person. W/m2 is the unit of the sound intensity used in Physics, which represents ‘watt per square metre’. For example: Given that I  103 W/m2. ∴ Loudness of sound  10 log dB  10 log 109 dB  10(9) dB  90 dB

5.5 Applications of Logarithms Sound intensity of 1 W/m2 is large enough to cause damage to our audition (hearing): Loudness of sound  10 log dB  10 log 1012 dB  10(12) dB  120 dB which is about the loudness of airplane’s engine. Loudness Example 20 dB Camera shutter 30 dB A silent park 40 dB A silent class 50 dB An office with typical sound 60 dB A conversation between two people one metre apart 80 dB MTR platform 100 dB Motor car’s horn 120 dB Plane’s engine

Example 5.17T 5.5 Applications of Logarithms Solution: If one person makes noise of 80 dB and another makes noise of 100 dB, then what is the ratio of the sound intensities made by the two people? Solution: Let I80 and I100 be the sound intensities made by the two people respectively. We can express each of the sound intensities I80 and I100 in terms of I0. ∴ I80 : I100  108I0 : 1010I0  1 : 100

5.5 Applications of Logarithms (b) Richter Scale The Richter scale R is a scale used to measure the magnitude of an earthquake: log E  4.8  1.5R where E is the energy released from an earthquake, measured in joules (J). The Richter scale was developed by an American scientist, Charles Richter. Remarks: Examples of serious earthquakes on Earth:  date: May 12, 2008 magnitude: 8.0 location: Sichuan province of China  date: Dec 26, 2004 magnitude: 9.0 location: Indian Ocean  date: May 22, 1960 magnitude: 9.5 location: Chile

Example 5.18T 5.5 Applications of Logarithms Solution: The most serious earthquake occurred in China was the Tangshan earthquake, which occurred on July 28, 1976. It was recorded as having the magnitude of 8.7 on the Richter scale. Compare the energy released by that earthquake with Taiwan’s 9-21 earthquake with a magnitude of 7.3 on the Richter scale in 1999. Solution: You may notice that for earthquakes with a difference in magnitude of 1.4 on the Richter scale, their energy released is almost 125 times greater. Since log E  4.8  1.5R, E  104.8  1.5R.  101.5(8.7 – 7.3)  102.1  126 ∴ The energy released by the Tangshan earthquake was 126 times that of Taiwan’s earthquake.

Chapter Summary 5.1 Rational Indices Laws of Indices For a, b  0, 1. am  an  am  n 2. am  an  am  n 3. (am)n  amn 4. (ab)m  ambm 5. 6. 7.

Chapter Summary 5.2 Logarithmic Functions If y  ax, then loga y  x, where a  0 and a  1. For base 10, we may write log y instead of log10 y. Properties of Logarithm For any positive numbers a, b, M and N with a, b  1, 1. loga a  1 2. loga 1  0 3. loga (MN)  loga M  loga N 4. loga  loga M  loga N 5. loga M n  n loga M 6. loga M

Chapter Summary 5.3 Using Logarithms to Solve Equations By using the properties of logarithm, we can solve the logarithmic equations and exponential equations.

Chapter Summary 5.4 Graphs of Exponential and Logarithmic Functions 1. The graphs of y  ax and y  are reflectionally symmetric about the y-axis. 2. The graphs of y  loga x and y  log1 x are reflectionally symmetric a about the x-axis. 3. The graphs of y  ax and y  loga x are symmetric about the line y  x.

Chapter Summary 5.5 Applications of Logarithms Daily-life applications of logarithms: 1. Measurement of loudness of sound in decibels (dB) 2. Measurement of magnitude of an earthquake on the Richter scale

Follow-up 5.1 5.1 Rational Indices Solution: B. Rational Indices Simplify , where b  0 and express the answer with a positive index. Solution: First express the inner radical with a rational index. Then use bm  bn  bmn to simplify the expression.

Follow-up 5.2 5.1 Rational Indices Solution: B. Rational Indices Evaluate . Solution:

Follow-up 5.3 5.1 Rational Indices Solution: B. Rational Indices Simplify . Solution:

Follow-up 5.4 5.1 Rational Indices Solution: D. Using the Law of Indices to Solve Equations Follow-up 5.4 Solve the equation for x  0. Solution: Change the equation into the form first.

Follow-up 5.5 5.1 Rational Indices Solution: D. Using the Law of Indices to Solve Equations Follow-up 5.5 Solve 3x  2  3x  270. Solution: 3x  2  3x  270 9(3x)  3x  270 Take out the common factor 3x first. 3x(9  1)  270 10 (3x)  270 3x  27  33 x  3

Follow-up 5.6 5.2 Logarithmic Functions Solution: B. Basic Properties of Common Logarithm Follow-up 5.6 Evaluate the following expressions. (a) log 4  log 5 (b) log 2  log 4  log 5 (c) Solution: (a) (c) (b) log 2  log 4  log 5

Follow-up 5.7 5.2 Logarithmic Functions Solution: B. Basic Properties of Common Logarithm Follow-up 5.7 Simplify , where x  0. Solution:

Follow-up 5.8 5.2 Logarithmic Functions Solution: B. Basic Properties of Common Logarithm Follow-up 5.8 If log 2  a and log 3  b, express the following in terms of a and b. (a) log 210 (b) log Solution: (b) log  log (a) log 210  10 log 2  10a  log 30  log (10  3)  (log 10  log 3) 

Follow-up 5.9 5.2 Logarithmic Functions Solution: C. Other Types of Logarithmic Functions Follow-up 5.9 Evaluate 5 log4 2  log4 2. Solution: 5 log4 2  log4 2  6 log4 2  6 log4 Express the number 2 in log4 2 in terms of the base 4.  6 log4 4  3

Follow-up 5.10 5.2 Logarithmic Functions Solution: D. Change of Base Formula Follow-up 5.10 Solve the equation logx 9  1.5. (Give the answer correct to 3 significant figures.) Solution: logx 9  1.5  1.5 log x   0.63616 x  4.33 (cor. to 3 sig. fig.)

Follow-up 5.11 5.2 Logarithmic Functions Solution: D. Change of Base Formula Follow-up 5.11 Show that log27 x  log3 x, where x  0. Solution: log27 x

Follow-up 5.12 5.3 Using Logarithms to Solve Equations Solution: A. Logarithmic Equations Follow-up 5.12 Solve the equation . Solution:

Follow-up 5.13 5.3 Using Logarithms to Solve Equations Solution: A. Logarithmic Equations Follow-up 5.13 Solve the equation log (x2  2)  log 5  1. Solution: log (x2  2)  log 5  1 log 5(x2  2)  1 5(x2  2)  10 5x2  20 x2  4 x  2

Follow-up 5.14 5.3 Using Logarithms to Solve Equations Solution: B. Exponential Equations Follow-up 5.14 Solve the equation 4x  2  12x. (Give the answer correct to 2 decimal places.) Solution: 4x  2  12x log 4x  2  log 12x (x  2) log 4  x log 12 x(log 4  log 12)  2 log 4 x   2.52 (cor. to 2 d. p.)

Follow-up 5.15 5.4 Graphs of Exponential and Logarithmic Functions D. Transformations on the Graphs of Exponential and Logarithmic Functions Follow-up 5.15 The following figure shows the graph of y  log2 x. Use the graph to sketch the graphs of the following functions: (a) y  log2 x (b) y  log2 (x – 2) Solution: (a) Since y  log2 x  log2 x  log2 y  log2  x 1 8  log2 x – 3 ∴ the graph of y  log2 x is obtained by translating the graph of y  log2 x three units downwards.

Follow-up 5.15 5.4 Graphs of Exponential and Logarithmic Functions D. Transformations on the Graphs of Exponential and Logarithmic Functions Follow-up 5.15 The following figure shows the graph of y  log2 x. Use the graph to sketch the graphs of the following functions: (a) y  log2 x (b) y  log2 (x – 2) Solution: (b) Let f (x)  log2 x, g(x)  log2 (x – 2). y  log2 (x  2) ∵ g(x)  f (x – 2) ∴ the graph of y  log2 (x – 2) is obtained by translating the graph of y  log2 x two units to the right.

Follow-up 5.16 5.4 Graphs of Exponential and Logarithmic Functions D. Transformations on the Graphs of Exponential and Logarithmic Functions Follow-up 5.16 The following figure shows the graph of y  2x. Use the graph to sketch the graphs of the following functions: (a) y  2x – 2 (b) y  1 – 2x Solution: (a) Let f (x)  2x, y  2x  2 g(x)  2x – 2. ∵ g(x)  f (x) – 2 ∴ the graph of y  2x – 2 is obtained by translating the graph of y  2x two units downwards.

Follow-up 5.16 5.4 Graphs of Exponential and Logarithmic Functions D. Transformations on the Graphs of Exponential and Logarithmic Functions Follow-up 5.16 The following figure shows the graph of y  2x. Use the graph to sketch the graphs of the following functions: (a) y  2x – 2 (b) y  1 – 2x Solution: (b) Let f (x)  2x, h(x)  1 – 2x. ∵ h(x)  1 – f (x) y  1  2x ∴ the graph of y  1 – 2x is obtained by reflecting the graph of y  2x about the x-axis, then translating one unit upwards.

Follow-up 5.17 5.5 Applications of Logarithms Solution: In a classroom of S4A, the loudness of the sound is 70 dB. In the classroom of S4B, the sound intensity is 10 times that of S4A. Find the loudness of sound produces by S4B. Solution: Let I4A be the sound intensity of S4A. Then . Loudness of sound produces by S4B

Follow-up 5.18 5.5 Applications of Logarithms Solution: Compare the energy released by the Chilean Quake with a magnitude of 9.5 on the Richter scale with Taiwan’s 9-21 earthquake with a magnitude of 7.3 on the Richter scale in 1999. Solution: Since log E  4.8  1.5R, E  104.8  1.5R.  101.5(9.5 – 7.3)  103.3  1995 ∴ The energy released by the Chilean Quake was 1995 times that of Taiwan’s earthquake.