3-2 Logarithmic Functions
Use the graph of f (x) to describe the transformation that results in the graph of g(x). Then sketch the
graphs of f (x) and g(x).
35. f (x) = log3 x; g(x) = log3 (x – 1)
SOLUTION: Find the relation of g(x) to f (x). g(x) = f (x − 1), so the graph of g(x) is the graph of f (x) translated 1 unit to the right.
36. f (x) = log x; g(x) = log 2x
SOLUTION: Find the relation of g(x) to f (x). g(x) = f (2x), so the graph of g(x) is the graph of f (x) compressed horizontally by a
factor of 2.
37. f (x) = ln x; g(x) = 0.5 ln x
SOLUTION: Find the relation of g(x) to f (x). g(x) = 0.5 · f (x), so the graph of g(x) is the graph of f (x) compressed vertically by a
factor of 0.5.
38. f (x) = log x; g(x) = –log (x – 2)
SOLUTION: Find the relation of g(x) to f (x). g(x) = −f (x − 2), so the graph of g(x) is the graph of f (x) reflected in the x-axis and
translated 2 units to the right.
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3-2 Logarithmic Functions
38. f (x) = log x; g(x) = –log (x – 2)
SOLUTION: Find the relation of g(x) to f (x). g(x) = −f (x − 2), so the graph of g(x) is the graph of f (x) reflected in the x-axis and
translated 2 units to the right.
39. f (x) = ln x; g(x) = 3 ln (x) + 1
SOLUTION: Find the relation of g(x) to f (x). g(x) = 3 · f (x) + 1, so the graph of g(x) is the graph of f (x) expanded vertically by a
factor of 3 and translated 1 unit up.
40. f (x) = log x; g(x) = –2 log x + 5
SOLUTION: Find the relation of g(x) to f (x). g(x) = −2 · f (x) + 5, so the graph of g(x) is the graph of f (x) reflected in the x-axis,
expanded vertically by a factor of 2, and translated 5 units up.
41. f (x) = ln x; g(x) = ln (–x)
SOLUTION: Find the relation of g(x) to f (x). g(x) = f (−x), so the graph of g(x) is the graph of f (x) reflected in the y-axis.
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3-2 Logarithmic Functions
41. f (x) = ln x; g(x) = ln (–x)
SOLUTION: Find the relation of g(x) to f (x). g(x) = f (−x), so the graph of g(x) is the graph of f (x) reflected in the y-axis.
42. INVESTING The annual growth rate for an investment can be found using r = ln , where r is the annual
growth rate, t is time in years, P is the present value, and P0 is the original investment. An investment of $10,000 was made in 2002 and had a value of $15,000 in 2009. What was the average annual growth rate of the investment?
SOLUTION: t = 7 years
P = 15,000
P0 = 10,000
Determine the domain, range, x-intercept, and vertical asymptote of each function.
43. y = log (x + 7)
SOLUTION: y = log (x + 7)
The log of zero or a negative number is undefined and x + 7 ≤ 0 when x ≤ −7, so D = {x | x > –7, x
asymptote is at x = −7.
}. The
The range of the logarithmic function is R = {y | y
}.
log 1 = 0
y = 0 when x + 7 = 1 → x-intercept: −6
44. y = log x – 1
SOLUTION: y = log x – 1
The log of zero or a negative number is undefined, so D = {x | x > 0, x
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The range of the logarithmic function is R = {y | y
}. The asymptote is at x = 0, the y-axis.
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}.
3-2
log 1 = 0
Functions
yLogarithmic
= 0 when x + 7 = 1 → x-intercept:
−6
44. y = log x – 1
SOLUTION: y = log x – 1
The log of zero or a negative number is undefined, so D = {x | x > 0, x
}. The asymptote is at x = 0, the y-axis.
The range of the logarithmic function is R = {y | y
}.
Find x when y = 0.
x-intercept: 10
45. y = ln (x – 3)
SOLUTION: y = ln (x – 3)
The ln of zero or a negative number is undefined and x − 3 ≤ 0 when x ≤ 3, so D = {x | x > 3, x
asymptote is at x = 3.
}. The
The range of the logarithmic function is R = {y | y
Find x when y = 0.
}.
x-intercept: 4
46. SOLUTION: The ln of zero or a negative number is undefined and x +
The asymptote is at x =
≤ 0 when x ≤ , so
.
The range of the logarithmic function is R = {y | y
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Find x when y = 0.
}.
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3-2 Logarithmic Functions
x-intercept: 4
46. SOLUTION: The ln of zero or a negative number is undefined and x +
The asymptote is at x =
≤ 0 when x ≤ , so
.
The range of the logarithmic function is R = {y | y
Find x when y = 0.
}.
x-intercept: 19.84
Find the inverse of each equation.
47. y = e3x
SOLUTION: 49. y = 4e2x
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3-2 Logarithmic Functions
49. y = 4e2x
SOLUTION: 51. y = 20x
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