Math Lesson­2.notebook
January 20, 2014
7.6 Solve Exponential & Log Equations
Day 2
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Logarithmic Equations: equations that involve logarithms of variable expressions.
Property of Equality for Logarithmic Equations: If b, x, and y are positive numbers with b ≠ 1, then logb x = logb y if and only if x = y.
Ex. 4: Solve
a) log5 (4x ­ 7) = log5 (x + 5)
b) ln (4x ­ 7) = ln (x + 11)
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Math Lesson­2.notebook
Ex. 5: Exponentiate/Solve a) log4 (5x ­ 1) = 3
January 20, 2014
b) log7 (2x + 1) = 2
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Math Lesson­2.notebook
January 20, 2014
Ex. 7:
The apparent magnitude of a star is a measure of the brightness of the star as it appears to observers on Earth. The apparent magnitude M of the dimmest star that can be seen with a telescope is given by the function
M = 5 log D + 2
where D is the diameter (in mm) of the telescope's objective lens. If a telescope can reveal stars with a magnitude of 12, what is the diameter of its objective lens?
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Math Lesson­2.notebook
January 20, 2014
Name
Jan. 20, 2014
Hour
p.519 #8­14e, 28­34e, 54, 61
Solve: Ex. 1
8. 33x­7 = 8112­3x
10. 103x­10 = (1/100)6x­1
Solve: Ex. 2
12. 8x = 20
14. 73x = 18
Solve & check for extraneous solutions. Ex. 4
28. log (12x ­ 11) = log (3x + 13)
30. log6 (3x ­ 10) = log6 (14 ­ 5x)
Solve & check for extraneous solutions. Ex. 5
32. log4 x = ­1
34. 1/3 log5 12x = 2
54. Ex. 3 You are cooking beef stew. When you take the beef stew off the stove, it has a temperature of 2000F. The room temperature is 750F and the cooling rate of the beef stew is r = 0.054. How long (in minutes) will it take to cool the beef stew to a temperature of 1000F?
61. Ex. 7
You plant a sunflower seedling in your garden. The seedling's height h (in cm) after t weeks can be modeled by the logistic function:
h(t) = 256 ­0.65t
1 + 13e
Find the time it takes the sunflower seedling to reach a height of 200 cm.
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Math Lesson­2.notebook
January 20, 2014
p.519 #8­14e, 28­34e, 54, 61
Solve: Ex. 1
10. 103x­10 = (1/100)6x­1
103x­10 = (10­2)6x­1
103x­10 = 10­12x+2
3x ­ 10 = ­12x + 2
+12x +10+12x +10
15x = 12
15 15
x = 0.8
Solve: Ex. 2
14. 73x = 18
log7 73x = log7 18
3x = log 18
log 7
3x = 1.485
3 3
x = 0.495
Solve & check for extraneous solutions. Ex. 4­5
34. 1/3 log5 12x = 2
28. log (12x ­ 11) = log (3x + 13)
3*1/3 log5 12x = 2*3
12x ­ 11 = 3x + 13
log5 12x = 6
­3x +11 ­3x +11
5log5 12x = 52 9x = 24
12x = 25
9 9
12 12
x = 2.667
x = 2.083
54. Ex. 3 You are cooking beef stew. When you take the beef stew off the stove, it has a temperature of 2000F. The room temperature is 75
and the cooling rate of the beef stew is r = 0.054. How long (in minutes) will it take to cool the beef stew to a temperature of 100
100 = (200 ­ 75)e­0.054t + 75
25 = 125e­0.054t 0.2 = e­0.054t ln 0.2 = ln e­0.054t ­1.609 = ­0.054t
t = 29.804
It will take about 30 minutes for the beef stew to cool to a temperature of 1000F.
61. h(t) = 256 ­0.65t
1 + 13e
256 (1 + 13e­0.65t)
(1 + 13e­0.65t)200 = 1 + 13e­0.65t
(1 + 13e­0.65t)200 = 256
200 200
1 + 13e­0.65t = 1.28
13e­0.65t = 0.28
13
13
­0.65t
e
= 0.022
ln e­0.65t = ln 0.022
­0.065t = ­3.838
t = ??
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