Expert: Sherman D. - 3/19/2005

Question
solve the exponential equation algebraically. Approximate the result to three decimal places.

31. 4(3^x)=20
32. 2(5^x)=32
33. 2e^x=80
34. 4e^x=91
35. e^x-9=19
36. 6^x+10=47
37. 3^(2x)=80
38. 6^(5x)=3000
39. 5^(-t/2)=0.20
40. 4^(-3t)=0.10
41. 3^(x-1)=27
42. 2^(x-3)=32
43. 2^(3-x)=27
44. 8^(-2-x)=431
45. 8(10^[3x])=12
46. 5(10^[x-6])=7
47. 3(5^[x-1])=21
48. 8(3^[6-x])=40
49. e^(3x)=12
50. e^(2x)=50
51. 500e^-x=3000
52. 1000e^(-4x)=75
53. 7-2e^x=5
54. -14+3e^x=11
55. 6(2^[3x-1])-7=9
56. 8(4^[6-2x])+13=41
57. e^(2x)-4e^x-5=0
58. e^(2x)-5e^x+6=0
59. e^(2x)-3e^x-4=0
60. e^(2x)+9e^x+36=0
61. 500/ 100-e^(x/2)=20
62. 400/ 1+e^-x=350

True or false? rewrite each verbal statement as an equation. Then decide whether the staatement is true or false, justify your answer.

121. The logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers.

122. The logarithm of the sum of two numbers is equal to the product og the logarithms of the numbers

123. The logarithm of the difference of two numbers is equal to the difference of the logarithms of the numbers.

124. The logarithm of the quotient of two numbers is equal to the difference of the logarithm numbers.

Answer
31.)
4(3^x) = 20
3^x = 5
log(3)5 = x
x = (log(5))/(log(3))
x = 1.46

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32.)
2(5^x) = 32
5^x = 16
Using same thing above
x = 1.723

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33.)
2e^x = 80
e^x = 40
Inverse e^x both sides
x = ln(40)
x = 3.689

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34.)
4e^x = 91
e^x = (91/4)
same rule above
x = ln(91/4)
x = 3.125

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35.)
e^x - 9 = 19
If by this you mean
(e^x) - 9 = 19
then
e^x = 10
inverse e^x both sides
x = 2.303

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36.)
6^x + 10 = 47
6^x = 37
same rule as 31
x = 2.015

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37.)
3^(2x) = 80
log(3)80 = 2x
(log(80))/(log(3)) = 2x
(log(80))/(2log(3)) = x
x = (log(80))/(log(3^2))
x = (log(80))/(log(9))
x = 1.994

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38.)
6^(5x) = 3000
(log(3000))/(log(6)) = 5x
x = (log(3000))/(5log(6))
x = (log(3000))/(log(6^5))
x = (log(3000))/(log(7776))
x = .894

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39.)
5^(-t/2) = 0.20
(log(.20))/(log(5)) = (-t/2)
(2log(.20))/(log(5)) = -t
t = (-2log(.20))/(log(5))
t = (log(.20^(-2)))/(log(5))
t = (log((1/5)^-2)))/(log(5))
t = (log(25))/(log(5))
t = 2

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40.)
4^(-3t) = 0.10
(log(.10))/(log(4)) = -3t
(log(.10))/(-3log(4)) = t
t = (log(.10))/(log(4^-3))
t = (log(1/10))/(log(1/64))
t = -1/(log(1/64))
t = .554

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41.)
3^(x - 1) = 27
3^(x - 1) = 3^3
x - 1 = 3
x = 4

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42.)
2^(x - 3) = 32
2^(x - 3) = 2^5
x - 3 = 5
x = 8

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43.)
2^(3 - x) = 27
(log(27))/(log(2)) = 3 - x
-x = (log(27)/log(2)) - 3
x = (-log(27)/log(2)) + 3
x = -1.755

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44.)
8^(-2 - x) = 431
(log(431))/(log(8)) = -2 - x
-x = (log(431)/log(8)) + 2
x = (-log(431)/log(8)) - 2
x = -4.917

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45.)
8(10^[3x]) = 12
10^(3x) = (3/2)
(log(3/2)/log(10)) = 3x
(log(3/2)/1) = 3x
log(3/2) = 3x
(log(3/2))/3 = x
x = .059

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46.)
5(10^[x - 6]) = 7
10^(x - 6) = (7/5)
(log((7/5))/log(10)) = x - 6
log(7/5) = x - 6
log(7/5) + 6 = x
x = 6.146

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47.)
3(5^[x - 1]) = 21
5^(x - 1) = 7
(log(7)/log(5)) = x - 1
x = (log(7)/log(5)) + 1
x = 2.209

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48.)
8(3^[6 - x]) = 40
3^(6 - x) = 5
(log(5)/log(3)) = 6 - x
-x = (log(5)/log(3) - 6
x = (-log(5)/log(3)) + 6
x = 4.535

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49.)
e^(3x) = 12
inverse e^x both sides
3x = ln(12)
divide both sides by 3
x = (ln(12))/3
x = .828

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50.)
e^(2x) = 50
2x = ln(50)
x = (ln(50))/2
x = 1.956

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51.)
500e^(-x) = 3000
e^(-x) = 6
-x = ln(6)
x = -ln(6)
x = -1.792

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52.)
1000e^(-4x) = 75
e^(-4x) = (3/40)
-4x = ln(3/40)
x = (-ln(3/40))/4
x = .648

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53.)
7 - 2e^x = 5
-2e^x = -2
e^x = 1
x = ln(1)
x = 0

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54.)
-14 + 3e^x = 11
3e^x = 25
e^x = (25/3)
x = ln(25/3)
x = 2.120

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55.)
6(2^[3x - 1]) - 7 = 9
6(2^(3x - 1)) = 16
2^(3x - 1) = (8/3)
(log(8/3))/(log(2)) = 3x - 1
(log(8/3)/log(2)) + 1 = 3x
(log(8/3)/(3log(2))) + (1/3) = x
x = (log(8/3)/(log(2^3))) + (1/3)
x = (log(8/3)/log(8)) + (1/3)
x = .805

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56.)
8(4^[6 - 2x]) + 13 = 41
8(4^(6 - 2x)) = 28
4^(6 - 2x) = (7/2)
(log(7/2)/log(4)) = 6 - 2x
(log(7/2)/log(4)) - 6 = -2x
(log(7/2)/(-2log(4))) + 6 = x
x = (log(7/2)/log(4^-2)) + 3
x = (log(7/2)/log(1/16)) + 3
x = 2.548

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57 - 60, i can't remember or find a way to answer them. If you know the way, PLEASE let me know, because i know i use to know how, but i can't remember. Also so that i may help you and others like with these kind of problems.


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61.)
500/(100 - e^(x/2)) = 20
500/20 = 100 - e^(x/2)
25 = 100 - e^(x/2)
-75 = -e^(x/2)
75 = e^(x/2)
ln(75) = x/2
2ln(75) = x
x = 8.635

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62.)
400/(1 + e^(-x)) = 350
1 + e^(-x) = (400/350)
1 + e^(-x) = (40/35)
1 + e^(-x) = 8/7
e^(-x) = (1/7)
-x = ln(1/7)
x = -ln(1/7)
x = 1.946

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True or false? rewrite each verbal statement as an equation. Then decide whether the statement is true or false, justify your answer.

121.) The logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers.

log(xy) = log(x) + log(y)
TRUE

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122. The logarithm of the sum of two numbers is equal to the product og the logarithms of the numbers

log(x + y) = log(x) + log(y)
False

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123. The logarithm of the difference of two numbers is equal to the difference of the logarithms of the numbers.

log(x - y) = log(x) - log(y)
False

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124. The logarithm of the quotient of two numbers is equal to the difference of the logarithm numbers.
log(x/y) = log(x) - log(y)
TRUE