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Section 10.3 1. The graph ofy = log(x) appears to be the mirror image of the graph of y = lOxwith the line y = x as the mirror. 3. a. log (1000)= 3 because1000= 103 b. log (10)= 1because10= 101 C. log (1) = 0 because1 = 10° d. 10g ~ ( )= 100 -2 because ~ = 10-2 100 5. 10g(l)< log(3.2) < 10g(l0) o <log(3.2) <1 log (3.2) is between 0 and 1. 7. 6(IW' = 600 (lW' = 100 10g(lW' =log(102) Take the log on both sides and write 100 as 102 The log undoes the base 10 exponential 2x=2 x=1 9. a. log(600)= x 2.77815..x b. log G) = 1.5x 10gG) -=x 1.5 -.31808= x c. log(.2)= 3x + 1 log(.2)-1 = 3x log(.2)-1 =x 3 -.56632.. x 11. a. x = 102= 100 b. x = 103= 1000 c. x = 101= 10 148 Chapter 10: Exponential and Logarithmic Functions 13. Yl = 2.7, Y2 = log (X) We need a large enough value for Xmax so that log (x) > 2.7. log (1000) = 3 so we canuse the mendly value, Xmax = 940. Given that Yl = 2.7, we can set Ymin = 0 and Ymax = 5. Windows may vary: Xmin = 0, Xmax = 940, Ymin = 0, Ymax = 5 b. 250=500(~J Yl = 250,Y2 = 500(3/4)"X Windowsmay vary:Xmin= 0, Xmax= 4.7, Ymin= 0, Ymax= 600 I"'- Irahrstc:ti(lfl M=2,'t091t201! _Y=250 The itemis worthhalfof its initialvaluein . Irahrstc:ti(lfl M=501.1II723 _ Y=2.7 X"" 501.187 The answerchecksbecauselog(501.187) 2.7 "" 15. You should try to follow the steps to solving this exercise. Many teachers may prefer to leave this exercise for future courses. See Example 8 on page 407 for a similar problem. .5 = .9i From the definition of the common logarithm on page 404 of the text, x may be written as approximately 2.4 years. 19. a. Ao= 750,r = .038,n = 12 SubstitutingforAo,r, n in the general compound formula, A(t) = (1 + ~ r 121 A(t) = 750 ( 1+ .~~8 ) 121 10Iog(x) . Applythisto .5 and .97in the equation to get 101og(.5) = ( 1 01og(.97) r ( Windows may vary: Xmin = 0, Xmax = 47, Ymin = 0, Ymax = 2000 101og(.5) = 10/.1og(.97) Because this is an equation, the left side must equal the right side. Because we have the same base on both sides of the equation, the powers must be equal. 10g(.5)=t *log(.97) log(.5) -=t 10g(.97) For an approximate solution, evaluate the expression for t on a calculator. t"" 22.76 Skills and Review 10.3 1 3 17. a. Yo=500, b=l--=-,k= 1 4 4 Substituting for Yo, b, k into the general exponential function y = yobb:,we have y=500GJ ) b. 1500= 750 1 + .~~8 Yl = 1500,Y2= 750(1+ .038/l2)"(12X) Irahrstc:ti(lfl M= 111.2':;95 B1 _ Y=15 00 ..............The account is worth $1,500 in approximately 18.3 years. 21. y = a,;; find a ifx = 5 andy = 175. 175=a*(5)2 175 = 25a 7=a 23. (c+3t=16 To undo the 4thpower on the left side raise to the power y. on both sides. Be sure to include the ::I: when taking an even root. 1/4 ( (c+W ) =:!:(16)1/4 c+3=:!:2 c = -3:!:2 c=-3+20r-3-2 c=-10rc=-5 @ Houghton Mifflin Company. All rights reserved. 25. 2 4 X -+-=- 3 5 30 Combine the fractions on left side, then use the cross multiplication property to solve for x. 2*5 4*3 x -+-=3 * 5 5 *3 30 10 12 X -+-=15 15 30 22 x 15 30 22 *30 = 15x 660 = 15x 44=x An alternative approach is to clear fractions by multiplying on both sides by 30. Although this method may save steps, it often leads to confusion when we simplify rational expressions and solve rational equations in Chapter II.