Logarithmic Functions
Digital Lesson
Definition: Logarithmic Function
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For x • 0, a • 0 and a ≠ 1,
y = loga x if and only if x = a y.
The function given by f (x) = loga x is called the logarithmic function with base a.
Every logarithmic equation has an equivalent exponential form:
y = loga x is equivalent to x = a y
A logarithmic function is the inverse function of an exponential function.
Exponential function: y = ax
Logarithmic function: y = logax is equivalent to x = ay
A logarithm is an exponent!
Examples: Write Equivalent Equations
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y = log2
Examples: Write the equivalent exponential equation
and solve for y.
1 = 5 y
y = log51
16 = 4y
y = log416
16 = 2y
y = log216
Solution
Equivalent Exponential Equation
Logarithmic Equation
16 = 24 → y = 4
→ y = –1
16 = 42 → y = 2
1 = 50 → y = 0
Common Logarithmic Function
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log10 –4
LOG (–) 4 ENTER
ERROR
no power of 10 gives a negative number
The base 10 logarithm function f (x) = log10 x is called the common logarithm function.
The LOG key on a calculator is used to obtain common logarithms.
Examples: Calculate the values using a calculator.
log10 100
log10 5
Function Value
Keystrokes
Display
LOG 100 ENTER
2
LOG 5 ENTER
0.6989700
log10
– 0.3979400
LOG ( 2 5 ) ENTER
Properties of Logarithms
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Examples:
log6 6 = 1
Simplify: log3 35
log3 35 = 5
Simplify: 7log79
7log79 = 9
Properties of Logarithms
1. loga 1 = 0 because a0 = 1.
2. loga a = 1 because a1 = a.
4. If loga x = loga y, then x = y. One-to-One Property
3. loga ax = x and alogax = x Inverse Properties
Solve for x: log6 6 = x
x = 1
Property 2
Property 3
Property 3
Example: Graph f(x) = log2 x
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x
y
4
4
Example: Graph f (x) = log2 x
Since the logarithm function is the inverse of the exponential function of the same base, its graph is the reflection of the exponential function in the line y = x.
8
3
4
2
2
1
1
0
–1
–2
2x
x
y = log2 x
y = x
y = 2x
Example: f(x) = log0 x
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Example: Graph the common logarithm function f(x) = log10 x.
1
0.602
0.301
0
–1
–2
f(x) = log10 x
10
4
2
1
x
y
x
5
–5
f(x) = log10 x
Graphing Utility: f(x) = log0 x
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Graphing Utility: Sketch the graphs of
f(x) = log10 x and f(x) = log10 (x – 1).
–0.5
4
1
–2
x = 1
f(x) = log10 (x – 1)
(1, 0)
(2, 0)
Graphs of Logarithmic Functions
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The graphs of logarithmic functions are similar for different values of a. f(x) = loga x (a • 1)
3. x-intercept: (1, 0)
6. Continuous
7. One-to-one
8. Reflection of y = a x in y = x
1. Domain:
2. Range:
4. Vertical asymptote: x = 0
Graph of f (x) = loga x, a • 1
x
y
y = x
y = loga x
y = a x
domain
range
y-axis
vertical
asymptote
(1, 0)
5. Increasing on
Natural Logarithmic Function
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The function defined by �f(x) = loge x = ln x
is called the natural �logarithmic function.
Use a calculator to evaluate: ln 3, ln –2, ln 100
ln 3
ln (–2)
ln 100
Function Value
Keystrokes
Display
LN 3 ENTER
1.0986122
ERROR
LN (–) 2 ENTER
LN 100 ENTER
4.6051701
y = ln x
(x • 0, e 2.718281…)
y
x
5
–5
y = ln x is equivalent to e y = x
Properties of Natural Logarithms
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Properties of Natural Logarithms
1. ln 1 = 0 since e0 = 1.
2. ln e = 1 since e1 = e.
3. ln ex = x and eln x = x
4. If ln x = ln y, then x = y.
Examples: Simplify each expression.
Inverse Property
Inverse Property
Property 2
Property 1
Inverse Properties
One-to-One Property