Section 6.4 Solutions #1 - 11: Write the equation in logarithmic form. 1) 32 = 9 (write the word log, make the 3 the base, switch the 2 and 9) Answer: log3 9 = 2 3) 34 =81 (write the word log, make the 3 the base, switch the 4 and 81) Answer: log3 81 = 4 5) (write the word log, make the 3 the base, switch the -1 and 1/3) Answer: 7) ey = x (write the word log, make the e the base, switch the y and x, then rewrite with ln notation) loge x = y Answer: ln(x) = y 9) e3 = 20.09 (write the word log, make the e the base, switch the 3 and 20.09, then rewrite with ln notation) Loge20.09 = 3 Answer: ln(20.09) = 3 11) e1 = 2.72 (write the word log, make the e the base, switch the 1 and 2.72, then rewrite with ln notation) Loge2.72 = 1 Answer: ln(2.72) = 1 #12 - 26: Write the equation in exponential form. 13) log381=4 (scratch out log, leave three low, switch 81 and 4) Answer: 34 = 81 15) log264=6 (scratch out log, leave e low, switch 64 and 6) Asnwer: 26 = 64 17) log6 6=1 (scratch out log, leave 6 low, switch 6 and 1) Answer: 61 = 6 19) log x=3 (write with base 10, then scratch out log, leave 10 low, switch x and 3) Log10 x = 3 Answer: 103 = x 21) ln(x) = 1 (write with base e, then scratch out log leave e low, switch 1 and x) Logex = 1 Answer: e1 = x 23) ln(2x) = w (write with base e, then scratch out log leave e low, switch 2x and w) Loge2x = w Answer: ew = 2x 25) ln(e2) = 2 (write with base e, then scratch out log leave e low, switch e2 and 2) Loge(e2) = 2 Answer: e2 = e2 26) ln(e3) = 3 (write with base e, then scratch out log leave e low, switch e3 and 3) Loge(e3) = 3 Answer: e3 = e3 #27 - 56: Find the logarithm value without using a calculator. 27) log2 2 (asks 2 to what power is 2, 2x = 2) Answer: 1 29) log7 7 (asks 7 to what power is 7, 7x = 7) Answer: 1 31) log3 1 (asks 3 to what power is 1, 3x = 1) Answer: 0 33) log4 1 (asks 4 to what power is 1, 4x = 1) Answer: 0 35) log4 64 (asks 4 to what power is 64, 4x = 64) Answer: 3 37) log5 5 (asks 5 to what power is 5, 5x = 5) Answer: 1 39) log 1 (asks 10 to what power is 1, 10x = 1) Answer: 0 41) log 100 (asks 10 to what power is 100, 10x = 100) Answer: 2 43) log2 128 (asks 2 to what power is 128, 2x = 128) Answer: 7 45) log2 23 (asks 2 to what power is 23, 2x = 23) Answer: 3 47) log5 56 (asks 5 to what power is 56, 5x = 56) Answer: 6 49) log4 45 (asks 4 to what power is 45, 4x = 45) Answer: 5 51) ln(e) (asks e to what power is e, ex = e) Answer: 1 53) ln(e3) (asks e to what power is e3, ex = e3) Answer: 3 55) ln(e-2) (asks e to what power is e-2, ex = e-2) Answer: -2 Section 6.4 Logarithmic Functions #57 - 65: Use a calculator to approximate the logarithms. Round to 4 decimal places. 57) log 6 ( I enter the problem exactly as it appears) Answer: .7782 59) ( I enter log(1/3) on my calculator) Answer:-.4771 61) log(5-3) (I enter log(5^-3) on my calculator) Answer:-2.0969 63) ln(7) (I type the problem as it appears) Answer: 1.9459 65) ln(24) (I enter Ln(2^4) on my calculator) Answer: 2.7726 #66-71 (graphs of common logarithmic functions) 67) y = log3 x a) Graph the logarithmic functions. First write the equation in exponential form, then create a table of values and plot the points. Write without log 3y = x Create a table of values, I will put the numbers 2,1,0,-1,-2 in the y column and solve for x. x 32 31 30 3-1 3-2 y 2 1 0 -1 -2 point (9,2) (3,1) (1,0) (1/3, -1) (1/9, -2) b) State the domain of each function. The graph is supposed to always be to the right of the y-axis. The graph is never supposed to touch the y-axis, so the domain should be x >0 Answer: 69) y = ln(x) a) Graph the logarithmic functions. First write the equation in exponential form, then create a table of values and plot the points. y = loge x (write so can see the base) ey = x (write without log) Create a table of values, I will put the numbers 2,1,0,-1,-2 in the y column and solve for x. x e2 e1 e0 e-1 e-2 y 2 1 0 -1 -2 point (7.39, 2) (2.72, 1) (1, 0) (.37, -1) (.14, -2) b) State the domain of each function. The graph is supposed to always be to the right of the y-axis. The graph is never supposed to touch the y-axis, so the domain should be x >0 Answer: 71) ⁄ a) Graph the logarithmic functions. First write the equation in exponential form, then create a table of values and plot the points. ( ) Create a table of values, I will put the numbers 2,1,0,-1,-2 in the y column and solve for x. x y 2 point (1/9, 2) ( ) 1 (1/3, 1) ( ) 0 (1, 0) ( ) -1 (3, -1) ( ) -2 (9,-2) ( ) b) State the domain of each function. The graph is supposed to always be to the right of the y-axis. The graph is never supposed to touch the y-axis, so the domain should be x >0 Answer: #72- 84: 73) y= log3(x+3) a) Graph the logarithmic functions. First write the equation in exponential form, then create a table of values and plot the points. 3y = x + 3 -3 + 3y = x Create a table of values, I will put the numbers 2,1,0,-1,-2 in the y column and solve for x. x -3 + 32 -3 + 31 -3 + 30 -3 + 3-1 -3 + 3-2 y 2 1 0 -1 -2 point (6, 2) (0,1) (-2,0) (-2.67, -1) (-.289, -2) b) State the domain of each function. The graph is supposed to always be to the right of the vertical line x = -3. The graph is never supposed to touch the line x = -3, so the domain should be x >-3 (the line x = -3 is drawn on the graph) Answer: c) describe the transformation the occurs from a common function Answer: same as graph of y=log3x, except shifted 3 units to left 75) y= log3(x – 2) a) Graph the logarithmic functions. First write the equation in exponential form, then create a table of values and plot the points. 3y = x-2 2 + 3y = x Create a table of values, I will put the numbers 2,1,0,-1,-2 in the y column and solve for x. x 2+32 2+31 2+30 2+3-1 2+3-2 y 2 1 0 -1 -2 point (11, 2) (5, 1) (3, 0) (2.33, -1) (2.11, -2) b) State the domain of each function. The graph is supposed to always be to the right of the vertical line x = 2. The graph is never supposed to touch the line x = 2, so the domain should be x >2 (the line x = 2 is drawn on the graph) Answer: c) describe the transformation the occurs from a common function Answer: same as graph of y=log3 x, except shifted 2 units to right 77) y = log3(x+1) a) Graph the logarithmic functions. First write the equation in exponential form, then create a table of values and plot the points. 3y = x + 1 -1 + 3y = x Create a table of values, I will put the numbers 2,1,0,-1,-2 in the y column and solve for x. x -1+32 -1+31 -1+30 -1+3-1 -1+3-2 y 2 1 0 -1 -2 point (8, 2) (2, 1) (0, 0) (-.67, -1) (-.89, -2) b) State the domain of each function. The graph is supposed to always be to the right of the vertical line x = -1. The graph is never supposed to touch the line x = -1, so the domain should be x >-1 (the line x = -1 is drawn on the graph) Answer: c) describe the transformation the occurs from a common function Answer: same as graph of y=log3 x, except shifted 1 unit to left 79) ⁄ a) Graph the logarithmic functions. First write the equation in exponential form, then create a table of values and plot the points. ( ) ( ) Create a table of values, I will put the numbers 2,1,0,-1,-2 in the y column and solve for x. x y 2 point (-3.75, 2) ( ) 1 (-3.5, 1) ( ) 0 (-3, 0) ( ) -1 (-2, -1) ( ) -2 (0, -2) ( ) b) State the domain of each function. The graph is supposed to always be to the right of the vertical line x = -4. The graph is never supposed to touch the line x = -4, so the domain should be x >-4 (the line x = -4 is drawn on the graph) Answer: c) describe the transformation the occurs from a common function Answer: same as graph of ⁄ , except shifted 4 units to left 81) y = ln(x) - 3 a) Graph the logarithmic functions. First write the equation in exponential form, then create a table of values and plot the points. y = loge(x) - 3 y + 3 = loge(x) ey+3 = x New table, since (-2) is the value of y that makes the exponent 0, I will put a (-2) in the middle of the ycolumn. x e0+3 e-1+3 e-2+3 e-3+3 e-4+3 y 0 -1 -2 -3 -4 point (20.09 , 0) (7.39 ,- 1) (2.72, -2) (1, -3) (.37, -4) b) State the domain of each function. The graph is supposed to always be to the right of the y-axis. The graph is never supposed to touch the y-axis, so the domain should be x >0 Answer: c) describe the transformation the occurs from a common function It’s harder to see this, but the (-3) shifts the graph down 3 units) Answer: same shape as y = ln(x), except shifted down 3 units. 83) y = ln(x) + 4 a) Graph the logarithmic functions. First write the equation in exponential form, then create a table of values and plot the points. y = loge(x) + 4 y – 4 = loge x ey – 4 = x New table, since 4 is the value of y that makes the exponent 0, I will put a 4 in the middle of the ycolumn. x e6-4 e5-4 e4-4 e3-4 ee-4 y 6 5 4 3 2 point (7.39, 6) (2.72,5) (1, 4) (.37, 3) (.14, 2) b) State the domain of each function. The graph is supposed to always be to the right of the y-axis. The graph is never supposed to touch the y-axis, so the domain should be x >0 Answer: c) describe the transformation the occurs from a common function It’s harder to see this, but the 4 shifts the graph up 4 units) Answer: same shape as y = ln(x), except shifted up 4 units.