Sect' 10'2 Logarithms and Logarithmic Functions

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Sect. 10.2 Logarithms and Logarithmic
Functions
Goal 1 Evaluate Logarithmic
Expressions Goal 2 Solve Logarithmic
Equations and Inequalities
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Logarithmic function with base a.
For x ? 0 and 0 ? a ? 1, y loga x if
and only if x a y.
A logarithm is an exponent!
A logarithmic function is the inverse of an
exponential function.
Exponential function y ax
Logarithmic function y logax is equivalent to
x ay
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Logarithms
y logax if and only if x ay
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Definition of a Logarithm

• A Logarithm, or log, is defined in terms of an
  exponential.
• If bx a, then logba x
• If 52 25 then log525 2
• log525 2 is read the log base 5 of 25 is 2.
• You might say the log is the exponent we put on
  5 to make 25

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Logarithms
Consider 72 49.
2 is the exponent of the power, to which 7 is
raised, to equal 49.
The logarithm of 49 to the base 7 is equal to 2
(log749 2).
Logarithmic form
Exponential notation
log749 2
72 49
In general If bx N,
then logbN x.
State in logarithmic form
State in exponential form
a) 63 216
log6216 3
a) log5125 3
53 125
b) log2128 7
27 128
b) 42 16
log416 2
2.5.5
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Logarithms
An logarithmic of x to the base b is defined by
log3 81 4 ? 34 81
log7 1 0 ? 70 1
log5 5 1 ? 51 5
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Evaluate
log3243
5
- 3
log366
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Evaluate
log992
2
x2 - 1
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Evaluate the Logarithms
Note log2128 log227 7
log327 log333 3
1. log2128
2. log327
log327 x 3x 27 3x 33
x 3
log2128 x 2x 128 2x 27
x 7
logaam m
3. log556
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4. log816
5. log81
log816 x 8x 16 23x 24
3x 4
log81 x 8x 1 8x 80 x 0
loga1 0
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The graphs of logarithmic functions are similar
for different values of a.
f(x) loga x (a ? 1)
y x
y a x
1. Domain Positive Real numbers
y log2 x
2. Range All Real Numbers
3. x-intercept (1, 0)
4. vertical asymptote y-axis
5. increasing
6. Continuous and one-to-one
7. reflection of y a x in y x
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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.
y 2x
y x
y log2 x
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Logarithmic Equation Equation that contains one
or more Logarithms.
Solve
x 27
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Solve the equation
n 16
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Solve each equation
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Write the equivalent exponential equation and
solve for y.
16 2y
y log216
16 24 ? y 4
16 4y
y log416
16 42 ? y 2
1 5y
y log51
1 50 ? y 0
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Rules for Logarithms

• Just as the rules for exponents let you easily
  rewrite a product, quotient, or power, the
  corresponding rules for logs allow you to rewrite
  the log of a product, the log of a quotient, or
  the log of a power.

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Multiplying with Exponents

• To multiply powers of the same base, keep the
  base and add the exponents.

Cant do anything about the y3 because its not
the same base.
Keep x, add exponents 7 5
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Dividing with Exponents

• To divide powers of the same base, keep the base
  and subtract the exponents.

Keep 5, subtract 12-4
Keep 7, subtract 10-6
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Powers with Exponents

• To raise a power to a power, keep the base and
  multiply the exponents.

This means t7t7t7 t777
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Evaluate the Logarithms
6.
7.
log4(log338)
x
log48 x 4x 8 22x 23 2x 3
2x 1
8.
23 8
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Logarithmic to Exponential Inequality
If b gt 1, x gt 0, and logbx gt y, then x gt by
If b gt 1, x gt 0, and logbx lt y, then 0 lt x lt by
Examples Log2x gt 3 x gt 23
Log3x lt 5 0 lt x lt 35
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Property of Equality for Logarithmic Functions
If b is a positive number other than 1, then
logbx logby if and only if x y
If log3x log37, then x 7
Extraneous Solutions The domain of logarithmic
functions does not include negative values.
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Solve the equation
log5 (x2 2) log5 x
x2 - 2 x
x2 - x - 2 0
(x 2)(x 1) 0
Check
x 2, -1
- 1 is an extraneous solution.
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Solve the equation
log4x2 log4(4x 3)
x2 4x - 3
x2 - 4x 3 0
(x 3)(x 1) 0
x 3, 1
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