5
Exponential and Logarithmic Functions
5.3
Logarithmic Functions
Definition 5.1 The logarithmic function to the base a, where a > 0 and
a 6= 1, is denoted by
y = loga x
(read as y is the logarithm to the base a of x) and is defined by
y = loga x ⇐⇒ x = ay .
The domain of the logarithmic function y = loga x is (0, ∞).
Example
• If y = log3 x, then
lent to 81 = 34 .
. For example, 4 = log3 81 is equiva-
• If y = log5 x, then
alent to 51 = 5−1 .
. For example, −1 = log5 ( 51 ) is equiv-
1. Express the equation in logarithmic form.
(a) 1.23 = m
(b) eb = 9
(c) a4 = 24
1
2. Express the equation in exponential form.
(a) loga 4 = 5
(b) loge b = −3
(c) log3 5 = c
3. Evaluate the following logarithmic expressions.
(a) log10 1000
(b) log10 0.1
(c) log16 4
(d) log2 16
(e) log3
1
27
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Properties of Logarithms
• loga 1 = 0
• loga a = 1
• loga ax = x
• aloga x = x
4. Evaluate the following logarithmic expressions.
(a) log5 1
(b) log5 58
(c) log5 5
(d) 5log5 12
Rule 5.2 The domain of the logarithmic function = the range of the expoand
nential function =
the range of the logarithmic function = the domain of the exponential function =
.
3
5. Sketch the graph of f (x) = log2 x and g(x) = log5 x.
6. Sketch the graph of f (x) = − log2 x and g(x) = log2 (−x).
7. Find the domain and sketch the graph of the function f (x) = 2+log5 x.
4
8. Find the domain and sketch the graph of the function
g(x) = log10 (x − 3).
9. Find the function of the form y = loga x whose graph is given.
10. Find the function of the form y = loga x whose graph is given.
5
Definition 5.3 If the base of the logarithmic function is the number e, then
we have the natural logarithmic function y = loge x = ln x.
If the base of a logarithmic function is the number 10, then we have the
common logarithm function y = log10 x = log x.
11. Use the definition of the logarithmic function to findx.
(a) log3 (4x − 7) = 2
(b) logx 64 = 2
(c) log7 (5x + 4) = 2
(d) log2 x = 2
(e) e2x = 5
(f) 103x = 6
6