1. Logarithmic Functions
Digital Lesson
Logarithmic Functions

2. Definition: Logarithmic Function
For x  0 and 0  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 logarithm is an exponent!
A logarithmic function is the inverse function of an exponential function.
Exponential function: y = ax
Logarithmic function: y = logax is equivalent to x = ay
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Definition: Logarithmic Function

3. Examples: Write Equivalent Equations
Examples: Write the equivalent exponential equation
and solve for y.
Solution
Equivalent Exponential Equation
Logarithmic Equation
y = log216
16 = 2y
16 = 24  y = 4
y = log2( )
= 2 y
= 2-1 y = –1
y = log416
16 = 4y
16 = 42  y = 2
y = log51
1 = 5 y
1 = 50  y = 0
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Examples: Write Equivalent Equations

4. Common Logarithmic Function
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.
Function Value
Keystrokes
Display
log10 100
LOG 100 ENTER 2
log10( )
LOG ( ) ENTER –
log10 5
LOG ENTER
log10 –4
LOG –4 ENTER
ERROR
no power of 10 gives a negative number
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Common Logarithmic Function

5. Properties of Logarithms
1. loga 1 = 0 since a0 = 1.
2. loga a = 1 since a1 = a.
3. loga ax = x and alogax = x inverse property
4. If loga x = loga y, then x = y. one-to-one property
Examples: Solve for x: log6 6 = x
log6 6 = 1 property 2 x = 1
Simplify: log3 35
log3 35 = 5 property 3
Simplify: 7log79
7log79 = 9 property 3
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Properties of Logarithms

6. Example: Graph the common logarithm function f(x) = log10 x.
0.602
0.301 –1 –2
f(x) = log10 x 10 4 2 x y x 5 –5
by calculator
f(x) = log10 x
(0, 1) x-intercept
x = 0 vertical asymptote
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Example: f(x) = log0 x

7. Graphs of Logarithmic Functions
The graphs of logarithmic functions are similar for different values of a f(x) = loga x (a  1)
y-axis
vertical
asymptote x y
Graph of f (x) = loga x (a  1)
y = a x
y = x
range
1. domain
y = log2 x
2. range
3. x-intercept (1, 0)
4. vertical asymptote
domain
x-intercept
(1, 0)
5. increasing
6. continuous
7. one-to-one
8. reflection of y = a x in y = x
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Graphs of Logarithmic Functions

8. Natural Logarithmic Functiony x 5 –5
y = ln x
The function defined by f(x) = loge x = ln x
(x  0, e )
is called the natural logarithm function.
y = ln x is equivalent to e y = x
Use a calculator to evaluate: ln 3, ln –2, ln 100
Function Value
Keystrokes
Display
ln 3
LN 3 ENTER
ln –2
LN –2 ENTER
ERROR
ln 100
LN 100 ENTER
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Natural Logarithmic Function

9. 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 inverse property
4. If ln x = ln y, then x = y. one-to-one property
Examples: Simplify each expression.
inverse property
inverse property
property 2
property 1
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Properties of Natural Logarithms