Section 4.3: Logarithmic Functions
Definition: If a > 0 and a 6= 1, then f (x) = ax is a one-to-one function. Therefore, it has an
inverse called the logarithmic function with base a, denoted by
f (x) = loga x.
The domain of f (x) = loga x is (0, ∞) and the range is R = (−∞, ∞). By definition,
y = loga x ⇐⇒ ay = x.
Moreover, since loga x and ax are inverse functions,
loga (ax ) = x = aloga x .
Example: Evaluate each expression.
(a) log3 27
(b) log6
1
36
(c) log8 4
1
Note: The most frequently used logarithms are the common logarithm, which has base 10,
and the natural logarithm which has the irrational number e as its base. There is special
notation for these logarithms.
log x = log10 x
ln x = loge x
Example: Evaluate each expression.
(a) log(0.01)
√
(b) ln( e)
Theorem: (Properties of Logarithms)
Suppose that x, y > 0 and n is a real number. Then
1. loga (xy) = loga x + loga y
x
= loga x − loga y
2. loga
y
3. loga (xn ) = n loga x
Example: Express each quantity as a single logarithm.
(a) log2 x + 5 log2 (x + 1) −
1
log2 (x − 1)
2
2
(b)
1
ln x − 4 ln(2x + 3)
3
Example: Solve each equation for x.
(a) 2x−5 = 3
(b) ln(2x − 1) = 3
(c) e3x−4 = 2
3
(d) ln x2 = 2 ln 4 − 4 ln 2
(e) ln(x + 6) + ln(x − 3) = ln 5 + ln 2
Note: Recall that the graph of y = f −1 (x) can be obtained by reflecting the graph of y = f (x)
about the line y = x. Therefore, the graph of y = loga x can be obtained from the graph of
y = ax . If 0 < a < 1, then loga x is decreasing and if a > 1, then loga x is increasing.
It follows that if a > 1, then
lim loga x = ∞
x→∞
and
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lim loga x = −∞.
x→0+
Example: Find the domain of ln(4 − x2 ).
Example: Evaluate each limit.
(a) lim+ ln(x − 3)
x→3
(b) lim log2 (x2 − x)
x→∞
(c) lim [ln(2x2 − 3x) − ln(x2 + 5x)]
x→∞
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Example: Find the inverse of each function.
(a) y = ln(x + 3)
(b) y =
1 + ex
1 − ex
Theorem: (Change of Base Formula)
Suppose that a > 0 and a 6= 1, then
loga x =
ln x
.
ln a
Example: Evaluate log5 21 correct to four decimal places.
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