Math 150 – Fall 2015
Section 5D
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Section 5D – Logarithmic Functions
Definition. Let a be any positive number not equal to 1. The logarithm of x to the
base a is y if and only if
ay = x.
The number y is denoted by
y = loga x
In other words, we say that y is the log of x to the base a, written y = loga x, if when
we raise a to the y th power we get x.
Note.
• The logarithm to a base a is the inverse function of ax .
• The expression loga x is read a “the log of x to the base x” or “the log to the base
a of x.
• Remember, to evaluate loga x we can write it as an exponential:
y = loga x
if and only if
ay = x
Example 1. Evaluate the following logarithms by writing an exponential equation.
(a) log10 10 =
because
(b) log100 10 =
because
(c) log√10 10 =
because
(d) log10 100 =
because
(e) log10 1000 =
because
(f) log5 25 =
because
(g) log2 64 =
because
(h) log4 64 =
because
(i) log8 64 =
because
Example 2. If (2.3)4 = x, what is a in loga x = 4?
Example 3. If the logarithm of x to the base 3 is 4, then x must equal what?
Example 4. Solve the equation log9 x = 3 for x.
Example 5. If loga 36 = 2, then what is a?
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Theorem. For a > 1, the plot of the logarithmic function with base a, written
y = loga x looks like the following.
• The domain of y = loga x is (0, ∞).
• The range of y = loga x is (−∞, ∞).
• When x = 1, we have loga 1 = 0 so the point (1, 0) is on the graph.
• When x = a, we have loga a = 1 so the point (a, 1) is on the graph.
• The function is increasing on (0, ∞).
• The function has a vertical asymptote at x = 0 and no horizontal asymptote.
For 0 < a < 1, the plot of the logarithmic function with base a, written y = loga x
looks like the following.
• The domain of y = loga x is (0, ∞), and the range is (−∞, ∞), the same as when
a > 1.
• When x = 1, we have loga 1 = 0 so the point (1, 0) is on the graph (the same as
when a > 1).
• When x = a, we have loga a = 1 so the point (a, 1) is on the graph. But now a is
between 0 and 1.
• When x = a1 , we have loga a1 = −1 so the point a1 , −1 is on the graph.
• The function is decreasing on (0, ∞).
• The function has a vertical asymptote at x = 0 and no horizontal asymptote.
Example 6. Graph the following functions. Give the domain, range, and intercepts of
the function.
(a) f (x) = log7 x
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Section 5D
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(b) g(x) = 2 log3 (x − 3) − 3
(c) h(x) = 3 log 13 (x + 1)
Properties of Logarithms
Theorem. Logarithms have the following properties (you need to memorize these):
1. The domain of loga x is (0, ∞) and its range is (−∞, ∞).
2. aloga x = x.
3. loga (ax ) = x
4. loga x = loga y if and only if x = y
5. loga 1 = 0
6. loga (xy) = loga x + loga y
7. loga (xy ) = y loga x
8. loga xy = loga x − loga y
Example 7. Simplify the expressions:
(a) 32 log9 5
(b) 74 log7 3
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Example 8. Simplify the following expressions:
(a) log4 7 − 3 log4 3 + 2 log4 5
(b) 2 log6 10 + 3 log6 3 − log6 75
Example 9. Solve the following expressions for x.
(a) 3 + log5 4 + 2 log5 x = 10
(b) 72x+3 = 11
Inverses
Theorem. The logarithmic function with base a and the exponential function with
base a are inverses since
loga x = y if and only if ay = x
Also, note the composition of the logarithmic function and exponential function equal
x:
aloga x = x and loga ax = x
Example 10. Graph the functions f (x) = 3x and g(x) = log3 x. Find the domain,
range, and intercepts of each function.
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Example 11. If a5 = 7.33, what is loga 7.33?
Example 12. If log5 13 = y, what is 5y ?
Natural Logarithm
Definition. The natural log function is the log function with base e. Instead of
writing loge x, we write ln x. The natural log function ln x is the inverse of the natural
exponential function ex . Therefore,
ln ex = x and eln x = x
Example 13. Graph ex and ln x. Give the domain, range, and intercepts of both
functions.
Example 14. Solve the equation e2x+7 = 11.
Example 15. Suppose that f (x) = aekx for some value of k and a. Suppose that
f (1) = 2 and f (3) = 1. Find a and k.
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Change of Base
Theorem. For any a, b > 0, we can rewrite the logarithmic function loga x using the
logarithmic function with base b using the formula
loga x =
logb x
logb a
Also, the exponential function ax can be rewritten for base b as
ax = bx logb a
Note. When we change base, we usually change to base e. So we can rewrite a logarithmic function and exponential function as
loga b =
ln b
ln a
and
ax = ex ln a
Note. Most calculators only have a button for log10 x and ln x. To evaluate a logarithmic function with any other base, we must use the change of base formula
Example 16. Use a calculator, to evaluate log13 25.
Example 17. Suppose that ln 2 = a, ln 3 = b, ln 5 = c, and ln 7 = d, and ln 11.
Evaluate and fully simplify the following.
(a) 11 log6 75
(b) 5 log55 24.