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Section 10.4 149 7. The relationship used to complete the table in Exercise 6 works the same way in reverse. If log (x) increases by 1, then x increases by a factor of 10. Or iflog (x) decreases by 1, then x decreases by a factor of 10. x log(x) .058 -1.2 .58 -.2 5.8 .8 58 1.8 580 2.8 9. It is hard to distinguish between amplitudes that vary so much. On the other hand, Richter magnitudes usually vary between 1 and 9 making it easier to see differences in earthquake intensity. Section 10.4 1. The domain off(x) = log (x) is the set of positive numbers. There is no number that we can put in the exponent of 10 that will give 0 or a negative number. 3. a. FindM(A)if A = I nun M(A) M(l) =10g(A) =log(l) = log(l 0°) =0 b. M(lOO)= 10g(100) = log(102) =2 11. The lower the pH, the more acidic the substance. For example, many plants prefer soil that is near neutral (pH = 7). When soils become acidic (pH < 7), gardeners apply lime (sodium hydroxide, pH = 12.4) to raise the pH of the soil. 13. a. pH and hydrogen ion concentration move in opposite directions. Lemon juice has a lower pH so it has the larger hydrogen ion concentration. b. The pH of lemon juice is 11 - 2 = 9 units lower, so its hydrogen ion concentration is 109 or 1,000,000,000 time larger. 15. The difference in intensities is 50 - 30 = 20 dB or 2 B. Because each bel represents multiplication by 10, 2 B represents multiplication by 102or 100. Thus, the noise from light traffic is 100 times louder than the sound from a soft whisper. c. M(lOO,OO) = 10g(100,000) = 10g(lOs) =5 Skills and Review 10.4 5. a. M(12,843)= log (12,843).. 4.1 17. a. 4x2+ 7 =43 b. M(240)= log (240).. 2.4 4x2 =36 c. M(351,060,000)= log (351,060,000)..8.5 N x2 =9 =:119 x=:13 Check 4x2 + 7 = 43 ? 4C3)2 + 7=43 ? 4*9+7=43 ? 36+7=43 43 = 43 @ Houghton MifflinCompany. All rights reserved. ? 4(3)2 + 7 =43 ? 4*9+7=43 ? 36+7=43 43 = 43 150 Chapter 10: Exponential and Logarithmic Functions b. 4-h -1 =11 4-h =12 -h=3 c. Again we could use the formula from part (a) to find A(2400) or recognize that 2400 years (-hf =32 x=9 Check 4-h-l=11 amount is .!.Ao=.!. * 3 kg = ~ kg or .375 kg. 8 8 8 4.;9-1~11 is three half-lives. .!.*.!. *.!. of the initial 222 23. fix) and g(x) are inverses of each other because f{g(x)) = x and g(f{x)) = x. fix) =x + 2 andg(x) =x - 2 f(g(x)) =f(x + 2) =(x-2)+2 Substitute 9 for x g(f(x)) = g(x + 2) =(x+2)-2 =x+2-2 =x ? 12 -1=11 11=11 19. a. 3(1Wx = 3000 (1WX = 1000 log(102X)= log(1000) 25. a. 2X2+ 6x 2x(x + 3) 10g(102x) 10g(103) b. x2y -16y y(x2 -16) y(x - 4)(x + 4) 2x=3 3 x = - =1.5 2 Check 3(lOh) =3000 ? 3(102'15)';'3000 Substitute 1.5 for x to middle -4xsplit thatthe sums to 3xterm, is -Ix3x. andThe 4x.factor pair of x2 -lx+4x-4 (x2 -lx)+(4x-4) x(x-l)+4(x-l) (x-I)(x+4) or (x+4)(x-l) ? 3(103)';'3000 ? 3 * 1000=3000 3000 = 3000 b. 410g(x) = - 4 10108(x) = 10-1 1 x=lO -1 =-=.1 10 Check 410g(x) =? -1 410g(1O -1)';' - 4 ? 4*Cl)=-4 c. ~ + 3x - 4 There is no GCF of all three terms and we can't use the difference of two squares, so try Substitute 10 -1 for x -4= -4 21. a. Ao= 3, b =.!., k = ~ ; substitutinginto A(t) 2 800 1 = Ao(b)''',we have A(t) = 3 2" ,/800 () . b. We could substitute 800 for t and find A(800) from the formula in (a), or recognize that 800 years is one half-life. In one half-life we have one-half the initial amount 1 1 3 - Au =- * 3 kg =- kg or 1.5 kg. 2 2 2