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Section 4.1 Logarithms and Their Properties
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For real numbers a, b, c with a > 0 and a 6= 1, suppose c = ab . Then a = c b . Then how do we
express b in terms of a and c?
Definition For a real number a with a > 0, a 6= 1, if c = ab , then we write b = loga c.
Example 1 Compute log2 64.
Example 2 Suppose loga 16 = 2. Determine a.
Example 3 loga 1 =
and loga a =
for all a with a > 0, x 6= 1.
Remark When we write y = loga x,
a. a > 0 and a 6= 1,
b. x > 0, and
c. y could be any real number.
Remark Recall the following properties: for every a with a > 0, a 6= 1 and for all real numbers x
and y,
a. ax · ay = ax+y
x
a
b.
= ax−y
y
a
x y
xy
c. (a ) = a
d. a0 = 1
e. a−x =
1
ax
f. If n is an integer
and m is a positive integer,
√
n
m
n
m
then a = a .
Theorem Let a > 0 and a 6= 1. Let x, y > 0.
a. loga xy = loga x + loga y.
b. loga
x
y
= loga x − loga y.
c. For every real numbers s(6= 0) and t, logas xt = st loga x. In particular, loga xt = t loga x.
d. y loga x = xloga y .
e. (Base Change Formula) If b > 0 and b 6= 1, then loga x =
logb x
.
logb a
In particular, loga b =
1
.
logb a
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Example 4 Let a > 0 and a 6= 1. Let x, y > 0. Express each of the following in terms of loga x and
loga y:
a. loga x4 y 3
b. loga
x2
y
c. loga ax
d. log 1 a2 x
a
Example 5 Let a > 0 and a 6= 1. Let x > 0. Simplify each of the following:
a. aloga x
b. a3 log
√
a
x
Definition
a. log x denotes log10 x. It is called the common logarithm of x.
b. ln x denotes loge x. It is called the natural logarithm of x.
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Example 6 Find x in each of the following equations, if possible.
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2
a. log 100 = x
d. log x =
b. log x = 3
e. log(−100) = x
c. log
√
10 = x
f. log x = −2
Example 7 Find x in each of the following equations.
√
3
a. ln e = x
c. ln
e=x
b. ln x = 2
d. ln e12 = x
Example 8 Solve the following equations.
a. 2x = 53.
c. ex = 11.
b. 3 · 5x = 20.
d. 102x−1 = 1000.
Example 9 Solve the equation log2 (x − 1) + log2 (x + 2) = 2.
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Section 4.2 Logarithms and Exponential Models
The log function is often useful when answering questions about exponential models.
Example 1 The population in a town has size 5000 at time t = 0, with t in years. Suppose
the population increases by 5% per year.
a. Find a formula for the population, P (t), at time t.
b. When would the population reach 10, 000?
Example 2 The voltage across a charged capacitor is given by V (t) = 5e−0.3t where t is in seconds.
a. What is the voltage after 3 seconds?
b. When will the voltage be 1?
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Section 4.3 The Logarithmic Function
Example 1 Sketch the graph of
a. y = log2 x.
b. y = log 1 x.
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Example 2 Determine
if y = 2x is invertible. If it is, find the formula for the inverse function. Do
x
the same for y = 31 .
Summary Let a > 0 and a 6= 1, then
a. The graph of y = loga x always passes through (1, 0), no matter what a is.
b. The domain of y = loga x is (0, ∞), no matter what a is.
c. The range of y = loga x is (−∞, ∞), no matter what a is.
d. If a > 1, then the graph of y = loga x is increasing and concave down.
e. If 0 < a < 1, then the graph of y = loga x is decreasing and concave up.
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Example 3 Sketch the graph of each of the following functions:
a. y = log2 (x − 4)
d. y = log 1002x
b. y = log2 x − 4
e. y = log2 (8x)
c. y = ln ex
f. y = log2
x
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