Logarithmic Functions
Joseph Lee
Metropolitan Community College
Joseph Lee
Logarithmic Functions
Logarithm
If b y = x, then logb x = y . The expression logb x is read “the
logarithm base b of x” or “log base b of x.”
Joseph Lee
Logarithmic Functions
Example 1.
Write the following exponential equations as logarithmic equations.
(a) 24 = 16
(b) 53 = 125
1
(c) 81 2 = 9
1
(d) 4−3 =
64
Joseph Lee
Logarithmic Functions
Example 1.
Write the following exponential equations as logarithmic equations.
(a) 24 = 16
(b) 53 = 125
1
(c) 81 2 = 9
1
(d) 4−3 =
64
Solution.
(a) log2 16 = 4
(b) log5 125 = 3
1
(c) log81 9 =
2
1
(d) log4
= −3
64
Joseph Lee
Logarithmic Functions
Example 2.
Write the following logarithmic equations as exponential equations.
(a) log10 1000 = 3
(b) log3 243 = 5
1
(c) log27 3 =
3
1
(d) −3 = log2
8
Joseph Lee
Logarithmic Functions
Example 2.
Write the following logarithmic equations as exponential equations.
(a) log10 1000 = 3
(b) log3 243 = 5
1
(c) log27 3 =
3
1
(d) −3 = log2
8
Solution.
(a) 103 = 1000
(b) 35 = 243
1
(c) 27 3 = 3
1
(d) 2−3 =
8
Joseph Lee
Logarithmic Functions
Example 3.
Evaluate the following logarithms.
(a) log5 25
(b) log10 10000
(c) log3 1
(d) log2 64
1
(e) log36
6
Joseph Lee
Logarithmic Functions
Example 3.
Evaluate the following logarithms.
(a) log5 25
(b) log10 10000
(c) log3 1
(d) log2 64
1
(e) log36
6
Solution.
(a) log5 25 = 2
(b) log10 10000 = 4
(c) log3 1 = 0
(d) log2 64 = 6
1
1
(e) log36 = −
6
2
Joseph Lee
Logarithmic Functions
Example 4.
Find the inverse function of f (x) = 2x .
Joseph Lee
Logarithmic Functions
Example 4.
Find the inverse function of f (x) = 2x .
Solution. Recall the inverse function interchanges the x and y
values.
f (x) = 2x
y = 2x
x = 2y
log2 x = y
f −1 (x) = log2 x
Joseph Lee
Logarithmic Functions
Example 5.a.
Graph the function f (x) = 2x and g (x) = log2 x.
Joseph Lee
Logarithmic Functions
Example 6.
Graph the function f (x) = log3 x. State the domain and range.
Joseph Lee
Logarithmic Functions
Example 6.
Graph the function f (x) = log3 x. State the domain and range.
x
f (x)
−2
−1
0
1
2
Joseph Lee
Logarithmic Functions
Natural and Common Logarithms
The natural logaritm of x is denoted ln x and ln x = loge x.
The common logaritm of x is denoted log x and log x = log10 x.
Joseph Lee
Logarithmic Functions
Example 7.
Graph the function f (x) = log x. State the domain and range.
Joseph Lee
Logarithmic Functions
Example 7.
Graph the function f (x) = log x. State the domain and range.
x
f (x)
−2
−1
0
1
2
Joseph Lee
Logarithmic Functions
Example 8.
Graph the function f (x) = ln x. State the domain and range.
Joseph Lee
Logarithmic Functions
Example 8.
Graph the function f (x) = ln x. State the domain and range.
x
f (x)
−2
−1
0
1
2
Joseph Lee
Logarithmic Functions
Example 9.
Graph the function f (x) = log2 (x − 3). State the domain and
range.
Joseph Lee
Logarithmic Functions
Example 10.
Graph the function f (x) = log2 x − 3. State the domain and range.
Joseph Lee
Logarithmic Functions