Outline Logarithm Functions Logarithms Properties
College Algebra & Trigonometry I 4.2 - Logarithm Functions
Math 1100 North Carolina Central University Math & C.S. Department Hicham Qasmi -
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Math 1100
College Algebra & Trigonometry I
Outline Logarithm Functions Logarithms Properties
1
Logarithm Functions Definition Properties Graphs
2
Logarithms Properties
Math 1100
College Algebra & Trigonometry I
Outline Logarithm Functions Logarithms Properties
1
Logarithm Functions Definition Properties Graphs
2
Logarithms Properties
Math 1100
College Algebra & Trigonometry I
Outline Logarithm Functions Logarithms Properties
Definition Properties Graphs
Outline
1
Logarithm Functions Definition Properties Graphs
2
Logarithms Properties
Math 1100
College Algebra & Trigonometry I
Outline Logarithm Functions Logarithms Properties
Definition Properties Graphs
Logarithm Functions
Definition For x > 0, b > 0 and b 6= 1, we define y = logb x as equivalent to the expression by = x The function f (x) = logb x is the logarithm function with base b.
Math 1100
College Algebra & Trigonometry I
Outline Logarithm Functions Logarithms Properties
Definition Properties Graphs
Logarithm Functions
Definition For x > 0, b > 0 and b 6= 1, we define y = logb x as equivalent to the expression by = x The function f (x) = logb x is the logarithm function with base b.
Math 1100
College Algebra & Trigonometry I
Outline Logarithm Functions Logarithms Properties
Definition Properties Graphs
Logarithm Functions Example 1
Write each equation in the equivalent exponential form: 2 = log2 x 3 = logb 64 log3 7 = y
Math 1100
College Algebra & Trigonometry I
Outline Logarithm Functions Logarithms Properties
Definition Properties Graphs
Logarithm Functions Example 2
Write each equation in the equivalent logarithm form: 1
122 = x
2
b3 = 8
3
ey = 9
Math 1100
College Algebra & Trigonometry I
Outline Logarithm Functions Logarithms Properties
Definition Properties Graphs
Logarithm Functions Example 3
Evaluate 1
log2 16
2
log3 9
3
log2 55
Math 1100
College Algebra & Trigonometry I
Outline Logarithm Functions Logarithms Properties
Definition Properties Graphs
Outline
1
Logarithm Functions Definition Properties Graphs
2
Logarithms Properties
Math 1100
College Algebra & Trigonometry I
Outline Logarithm Functions Logarithms Properties
Definition Properties Graphs
Basic Properties
Theorem 1
logb b = 1
2
logb 1 = 0
Math 1100
College Algebra & Trigonometry I
Outline Logarithm Functions Logarithms Properties
Definition Properties Graphs
Basic Properties
Theorem 1
logb b = 1
2
logb 1 = 0
Math 1100
College Algebra & Trigonometry I
Outline Logarithm Functions Logarithms Properties
Definition Properties Graphs
Example
Evaluate 1
log7 7
2
log5 1
Math 1100
College Algebra & Trigonometry I
Outline Logarithm Functions Logarithms Properties
Definition Properties Graphs
Inverse Properties
Theorem 1
logb bx = 1
2
logb 1 = 0
Math 1100
College Algebra & Trigonometry I
Outline Logarithm Functions Logarithms Properties
Definition Properties Graphs
Inverse Properties
Theorem 1
logb bx = 1
2
logb 1 = 0
Math 1100
College Algebra & Trigonometry I
Outline Logarithm Functions Logarithms Properties
Definition Properties Graphs
Inverse Properties
Theorem 1
logb bx = 1
2
logb 1 = 0
Math 1100
College Algebra & Trigonometry I
Outline Logarithm Functions Logarithms Properties
Definition Properties Graphs
Inverse Properties
Theorem 1
logb bx = 1
2
logb 1 = 0
Math 1100
College Algebra & Trigonometry I
Outline Logarithm Functions Logarithms Properties
Definition Properties Graphs
Example
Evaluate 1
log4 45
2
6log6 9
Math 1100
College Algebra & Trigonometry I
Outline Logarithm Functions Logarithms Properties
Definition Properties Graphs
Outline
1
Logarithm Functions Definition Properties Graphs
2
Logarithms Properties
Math 1100
College Algebra & Trigonometry I
Outline Logarithm Functions Logarithms Properties
Definition Properties Graphs
Example of Graph Graph f (x) = 2x and g(x) = log2 x in the same coordinate system. 5 4
y = 2x
3 2 1 0 −5 −4 −3 −2 −1 0 −1
1
2
3
4
5
−2 −3 −4 −5 Math 1100
College Algebra & Trigonometry I
Outline Logarithm Functions Logarithms Properties
Definition Properties Graphs
Example of Graph Graph f (x) = 2x and g(x) = log2 x in the same coordinate system. 5 4
y = 2x
reflection
3 2 1 0 −5 −4 −3 −2 −1 0 −1
1
2
3
4
5
−2 −3 −4 −5 Math 1100
College Algebra & Trigonometry I
Outline Logarithm Functions Logarithms Properties
Definition Properties Graphs
Example of Graph Graph f (x) = 2x and g(x) = log2 x in the same coordinate system. 5 4
y = 2x
reflection
3 2 1 0 −5 −4 −3 −2 −1 0 −1
y = log2 x 1
2
3
4
5
−2 −3 −4 −5 Math 1100
College Algebra & Trigonometry I
Outline Logarithm Functions Logarithms Properties
Definition Properties Graphs
Characteristics of the Graph of f (x) = logb x
1
The x-intercept is 1.
2
There is no y-intercept.
3
The y-axis (or x = 0) is a vertical asymptote.
4
If b > 1, the function is increasing. If 0 < b < 1, the function is decreasing.
5
The graph is smooth and continuous.
6
The domain is (0, ∞)
Math 1100
College Algebra & Trigonometry I
Outline Logarithm Functions Logarithms Properties
Definition Properties Graphs
Characteristics of the Graph of f (x) = logb x
1
The x-intercept is 1.
2
There is no y-intercept.
3
The y-axis (or x = 0) is a vertical asymptote.
4
If b > 1, the function is increasing. If 0 < b < 1, the function is decreasing.
5
The graph is smooth and continuous.
6
The domain is (0, ∞)
Math 1100
College Algebra & Trigonometry I
Outline Logarithm Functions Logarithms Properties
Definition Properties Graphs
Characteristics of the Graph of f (x) = logb x
1
The x-intercept is 1.
2
There is no y-intercept.
3
The y-axis (or x = 0) is a vertical asymptote.
4
If b > 1, the function is increasing. If 0 < b < 1, the function is decreasing.
5
The graph is smooth and continuous.
6
The domain is (0, ∞)
Math 1100
College Algebra & Trigonometry I
Outline Logarithm Functions Logarithms Properties
Definition Properties Graphs
Characteristics of the Graph of f (x) = logb x
1
The x-intercept is 1.
2
There is no y-intercept.
3
The y-axis (or x = 0) is a vertical asymptote.
4
If b > 1, the function is increasing. If 0 < b < 1, the function is decreasing.
5
The graph is smooth and continuous.
6
The domain is (0, ∞)
Math 1100
College Algebra & Trigonometry I
Outline Logarithm Functions Logarithms Properties
Definition Properties Graphs
Characteristics of the Graph of f (x) = logb x
1
The x-intercept is 1.
2
There is no y-intercept.
3
The y-axis (or x = 0) is a vertical asymptote.
4
If b > 1, the function is increasing. If 0 < b < 1, the function is decreasing.
5
The graph is smooth and continuous.
6
The domain is (0, ∞)
Math 1100
College Algebra & Trigonometry I
Outline Logarithm Functions Logarithms Properties
Definition Properties Graphs
Characteristics of the Graph of f (x) = logb x
1
The x-intercept is 1.
2
There is no y-intercept.
3
The y-axis (or x = 0) is a vertical asymptote.
4
If b > 1, the function is increasing. If 0 < b < 1, the function is decreasing.
5
The graph is smooth and continuous.
6
The domain is (0, ∞)
Math 1100
College Algebra & Trigonometry I
Outline Logarithm Functions Logarithms Properties
Definition Properties Graphs
Characteristics of the Graph of f (x) = logb x
1
The x-intercept is 1.
2
There is no y-intercept.
3
The y-axis (or x = 0) is a vertical asymptote.
4
If b > 1, the function is increasing. If 0 < b < 1, the function is decreasing.
5
The graph is smooth and continuous.
6
The domain is (0, ∞)
Math 1100
College Algebra & Trigonometry I
Outline Logarithm Functions Logarithms Properties
Definition Properties Graphs
Domain example
Example Find the domain of f (x) = log4 (x + 3)
Math 1100
College Algebra & Trigonometry I
Outline Logarithm Functions Logarithms Properties
Definition Properties Graphs
Common Logarithm
The common logarithm is the logarithm with base b = 10: log x = log10 x
Math 1100
College Algebra & Trigonometry I
Outline Logarithm Functions Logarithms Properties
Definition Properties Graphs
Natural Logarithm
The naturel logarithm is the logarithm with base b = e: ln x = loge x
Math 1100
College Algebra & Trigonometry I
Outline Logarithm Functions Logarithms Properties
Logarithms Properties
Theorem Let b, M, N and p be positive real numbers with b 6= 1. Then, 1
2
3
logb (MN) = logb M + logb N M logb ( ) = logb M − logb N N logb (M p ) = p · logb M
Math 1100
College Algebra & Trigonometry I
Outline Logarithm Functions Logarithms Properties
Logarithms Properties
Theorem Let b, M, N and p be positive real numbers with b 6= 1. Then, 1
2
3
logb (MN) = logb M + logb N M logb ( ) = logb M − logb N N logb (M p ) = p · logb M
Math 1100
College Algebra & Trigonometry I
Outline Logarithm Functions Logarithms Properties
Logarithms Properties
Theorem Let b, M, N and p be positive real numbers with b 6= 1. Then, 1
2
3
logb (MN) = logb M + logb N M logb ( ) = logb M − logb N N logb (M p ) = p · logb M
Math 1100
College Algebra & Trigonometry I