115.3 Assignment#3 Solutions-1 Sa 115.3 Assignment#3 Solutions sk ln x 2 − 1 loge x 2 − 1 = x −1 = loge 2 ln 2 (a) log2 x − 1 Solution: log2 (b) log3 (5x + 1) Solution: log3 (5x + 1) = 2 2 loge (5x + 1) ln(5x + 1) = loge 3 ln 3 log (x + 2) ln(x + 2) e = log10 x 2 − 1 = loge 10 ln 10 2 log 2x 2 − 1 ln 2x − 1 e = (d) log2 2x 2 − 1 Solution: log2 2x 2 − 1 = loge 2 ln 2 (c) log(x + 2) Solution: y 1.3-2 Sketch the graph of y = −(x − 2) + 1 without using a graphing calculator. 2 Solution: Starting with the graph of y = x 2 in black, we get the graph shifted 2 units to the right in blue: y = (x − 2)2 , that graph reﬂected about the x-axis in green: y = −(x − 2)2 , and ﬁnally that graph shifted up 1 unit in red: y = −(x − 2)2 + 1. PAT- ET RIÆ atc h e w ane n 2003 Doug MacLean 1.2-86 Write the following expressions in terms of base e. DEO sis iversitas Un 10 9 8 7 6 5 4 3 2 1 0 -1 1 2 3 4 -4-3-2-10 -2 -3 -4 -5 -6 -7 -8 -9 -10 x y Solution: Starting with the graph of y = ex in black, we get that reﬂected about the y-axis graph in red: y = e−x . sk DEO PAT- ET RIÆ atc h sis 1.3-12 Sketch the graph of y = ex without using a graphing calculator. iversitas Un Sa 10 9 8 7 6 5 4 3 2 1 0 -4-3-2-10 1 2 3 4 115.3 Assignment#3 Solutions-2 e w ane n 2003 Doug MacLean x 1.3-34 Find the following numbers on a number line that is on a logarithmic scale (base 10): 0.03,0.7,1,2,5,10,1.7,100,150, and 2000. Solution: 0.03 1 -2 10 2 3 4 5 6 7 891 -1 10 2 0.7 1 2 3 4 5 6 7 891 2 0 10 5 10 3 4 5 6 7 891 1 10 17 2 100150 3 4 5 6 7 891 2 10 2000 2 1 3 4 5 6 7 89 3 10 2 3 4 5 6 7 89 4 10 115.3 Assignment#3 Solutions-3 1.3-36 Find the following numbers on a number line that is on a logarithmic scale (base 10): Sa sk (a) 10−3 , 2 × 10−3 , 5 × 10−3 DEO PAT- ET RIÆ atc h sis iversitas Un e w ane n 2003 Doug MacLean (b) 10−1 , 2 × 10−1 , 5 × 10−1 (c) 102 , 2 × 102 , 2 × 102 (d) Using (a)-(c), how many units (on a logarithmic scale) is 2 × 10−3 from 10−3 (2 × 10−2 from 10−2 , 2 × 102 from 102 )? (e) Using (a)-(c), how many units (on a logarithmic scale) is 5 × 10−3 from 10−3 (5 × 10−2 from 10−2 , 5 × 102 from 102 )? Solution: (d) 1 , (e) 4 0.03 1 -2 10 2 3 4 5 6 7 891 -1 10 2 0.7 1 2 3 4 5 6 7 891 2 0 10 5 10 3 4 5 6 7 891 1 10 17 2 100150 3 4 5 6 7 891 2 10 2000 2 1 3 4 5 6 7 89 3 10 1.3-38 Both the La Plata river dolphin and the sperm whale are marine mammals having teeth. A La Plata river dolphin weighs between 30 and 50 kg, whereas a sperm whale weighs between 35,000 and 40,000 kg. How many orders of magnitude greater is the weight of the sperm whale? 40, 000 =3 Solution: log10 40 2 3 4 5 6 7 89 4 10 115.3 Assignment#3 Solutions-4 (x1 , y1 ) = (−1, 4), Sa 1.3-44 When log y is graphed as a function of x on log-linear paper, a straight line results. Graph straight lines, each given by two points, on a log-linear plot, and determine the functional relationship. sk (x2 , y2 ) = (2, 8) Since the graph is a straight line on log-linear graph paper, we have Y=mx+b for constants m and b which we must ﬁnd. At (x1 , y1 ) = (−1, 4) we have log 4 = m(−1) + b, and at (x2 , y2 ) = (2, 8) we have log 8 = m(2) + b, so on subtracting the ﬁrst equation from the second, we get log 84 log 8 − log 4 log 2 log 8 − log 4 = 3m or m = = = 0.100 3 3 3 and we have the equation Y = log 2 7 3 x+ 3 = 2 log 2 + log 2 3 = 7 3 log 2 0.702 log 2 7 x + log 2. 3 3 Exponentiating, we get y = 10Y = 10log y = 10 log 2 3 log 2 log 2 x 1 x 7 x 7 7 3 3 = 10 3 log 2 × 10 3 = 10log 2 × 10log 2 = 23 × 23 5.04 × 1.26x Y=log y 9 8 7 6 5 4 3 2 1 -2 -1 0 1 2 PAT- ET RIÆ atc h e w ane n 2003 Doug MacLean Solution: Next, we have b = log 4 + m = log 4 + DEO sis iversitas Un 3 x 115.3 Assignment#3 Solutions-5 y = 4 × 105x . Solution: Sa 1.3-48 Use a logarithmic transformation to ﬁnd a linear relationship between the given quantities and graph the resulting linear relationship on a log-linear plot: sk We have Y = log(4 × 10 ) = log 4 + 5x 9 8 7 6 5 4 3 2 1 -1 0 PAT- ET RIÆ atc h e w ane n 2003 Doug MacLean Y=log y 5x DEO sis iversitas Un 1 x 115.3 Assignment#3 Solutions-6 Solution: We have Y = log y = log 5, so y = 5 . y 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 Sa 1.3-56 When log y is graphed as a function of log x a straight line results. Graph straight lines, each given by two points on a log-log plot, and determine the functional relationship. (The original x-y coordinates are given.) (x1 , y1 ) = (3, 5), (x2 , y2 ) = (1, 5) sk DEO PAT- ET RIÆ atc h sis iversitas Un e w ane n 2003 Doug MacLean 115.3 Assignment#3 Solutions-7 y = 3x 2 Solution: Y = log y = log(3x 2 ) = log 3 + 2 log x = log 3 + 2X, where X = log x. y 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 Sa 1.3-60 Use a logarithmic transformation to ﬁnd a linear relationship between the given quantities and graph the resulting linear relationships on a log-log plot. sk DEO PAT- ET RIÆ atc h sis iversitas Un e w ane n 2003 Doug MacLean 115.3 Assignment#3 Solutions-8 g(s) = 1.8e−0.2s Solution: G = log g(s) = log 1.8e−0.2s = log 1.8 + log e−0.2s = log 1.8 − 0.2s log e = log 1.8 − (0.2 log e)s which we plot on log-linear paper: G=log g 9 8 7 6 5 4 3 2 1 -1 0 1 s Sa 1.3-68 Use a logarithmic transformation to ﬁnd a linear relationship between the given quantities and determine whether a log-log or a log-linear plot should be used to graph the resulting relationship sk DEO PAT- ET RIÆ atc h sis iversitas Un e w ane n 2003 Doug MacLean 115.3 Assignment#3 Solutions-9 Solution: L(c) = 1.7 × 102.3c log L(c) = log 1.7 × 102.3c = log 1.7 + log 102.3c = log 1.7 + 2.3c which we plot on log-linear paper: log L 9 8 7 6 5 4 3 2 1 -1 0 1 c Sa 1.3-72 Use a logarithmic transformation to ﬁnd a linear relationship between the given quantities and determine whether a log-log or a log-linear plot should be used to graph the resulting relationship sk DEO PAT- ET RIÆ atc h sis iversitas Un e w ane n 2003 Doug MacLean

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