Notes 6-3
60
Section 6-3 Logarithms
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Logarithm Functions
Transcript of Notes 6-3
Section 6-3L o g a r i t h m s
Warm-upSolve without a calculator.
a. 10a = .0001 b. 10a = .01 c. 10a = 1
d. 10a = 10 e. 10a = 100
f. 10a = 100,000,000,000 g. 10a = 0
Warm-upSolve without a calculator.
a. 10a = .0001 b. 10a = .01 c. 10a = 1
d. 10a = 10 e. 10a = 100
f. 10a = 100,000,000,000 g. 10a = 0
a = −4
Warm-upSolve without a calculator.
a. 10a = .0001 b. 10a = .01 c. 10a = 1
d. 10a = 10 e. 10a = 100
f. 10a = 100,000,000,000 g. 10a = 0
a = −2 a = −4
Warm-upSolve without a calculator.
a. 10a = .0001 b. 10a = .01 c. 10a = 1
d. 10a = 10 e. 10a = 100
f. 10a = 100,000,000,000 g. 10a = 0
a = −2 a = 0 a = −4
Warm-upSolve without a calculator.
a. 10a = .0001 b. 10a = .01 c. 10a = 1
d. 10a = 10 e. 10a = 100
f. 10a = 100,000,000,000 g. 10a = 0
a = −2 a = 0
a = 1
2
a = −4
Warm-upSolve without a calculator.
a. 10a = .0001 b. 10a = .01 c. 10a = 1
d. 10a = 10 e. 10a = 100
f. 10a = 100,000,000,000 g. 10a = 0
a = −2 a = 0
a = 1
2 a = 2
a = −4
Warm-upSolve without a calculator.
a. 10a = .0001 b. 10a = .01 c. 10a = 1
d. 10a = 10 e. 10a = 100
f. 10a = 100,000,000,000 g. 10a = 0
a = −2 a = 0
a = 1
2 a = 2
a = 11
a = −4
Warm-upSolve without a calculator.
a. 10a = .0001 b. 10a = .01 c. 10a = 1
d. 10a = 10 e. 10a = 100
f. 10a = 100,000,000,000 g. 10a = 0
a = −2 a = 0
a = 1
2 a = 2
a = 11 No solution
a = −4
Definition of Logarithm
Definition of Logarithm
Let b > 0 and b ≠ 1. Then y is the logarithm of x to the base b, written:
Definition of Logarithm
Let b > 0 and b ≠ 1. Then y is the logarithm of x to the base b, written:
y = log
bx IFF by = x
Definition of Logarithm
Let b > 0 and b ≠ 1. Then y is the logarithm of x to the base b, written:
What does this mean?
y = log
bx IFF by = x
Definition of Logarithm
Let b > 0 and b ≠ 1. Then y is the logarithm of x to the base b, written:
What does this mean?
y = log
bx IFF by = x
y = log
bx IFF by = x
Definition of Logarithm
Let b > 0 and b ≠ 1. Then y is the logarithm of x to the base b, written:
What does this mean?
y = log
bx IFF by = x
Base
y = log
bx IFF by = x
Definition of Logarithm
Let b > 0 and b ≠ 1. Then y is the logarithm of x to the base b, written:
What does this mean?
y = log
bx IFF by = x
Base
Exponent
y = log
bx IFF by = x
Example 1Evaluate.
a. log
616
b. log636
c. log
6365
Example 1Evaluate.
a. log
616
b. log636
c. log
6365
−1
Example 1Evaluate.
a. log
616
b. log636
c. log
6365
Why?
−1
Example 1Evaluate.
a. log
616
b. log636
c. log
6365
Why?
−1
6−1 = 1
6
Example 1Evaluate.
a. log
616
b. log636
c. log
6365
Why?
−1
6−1 = 1
6
2
Example 1Evaluate.
a. log
616
b. log636
c. log
6365
Why?
−1
6−1 = 1
6
2
Why?
Example 1Evaluate.
a. log
616
b. log636
c. log
6365
Why?
−1
6−1 = 1
6
2
Why?
62 = 36
Example 1Evaluate.
a. log
616
b. log636
c. log
6365
Why?
−1
6−1 = 1
6
2
Why?
62 = 36
25
Example 1Evaluate.
a. log
616
b. log636
c. log
6365
Why?
−1
6−1 = 1
6
2
Why?
62 = 36
25
Why?
Example 1Evaluate.
a. log
616
b. log636
c. log
6365
Why?
−1
6−1 = 1
6
2
Why?
62 = 36
25
Why?
625 = 365 = 625
Example 2Evaluate.
log
9243
Example 2Evaluate.
log
9243
92 = 81
Example 2Evaluate.
log
9243
92 = 81 9
3 = 729
Example 2Evaluate.
log
9243
92 = 81 9
3 = 729x is somewhere in between
Example 2Evaluate.
log
9243
92 = 81 9
3 = 729x is somewhere in between
What do we know about 243?
Example 2Evaluate.
log
9243
92 = 81 9
3 = 729x is somewhere in between
What do we know about 243?
2435 = 3
Example 2Evaluate.
log
9243
92 = 81 9
3 = 729x is somewhere in between
What do we know about 243?
2435 = 3 = 912
Example 2Evaluate.
log
9243
92 = 81 9
3 = 729x is somewhere in between
What do we know about 243?
Ok, what does that mean? 2435 = 3 = 9
12
Example 2Evaluate.
log
9243
92 = 81 9
3 = 729x is somewhere in between
What do we know about 243?
Ok, what does that mean? 2435 = 3 = 9
12
9
12( )5
= 243
Example 2Evaluate.
log
9243
92 = 81 9
3 = 729x is somewhere in between
What do we know about 243?
Ok, what does that mean? 2435 = 3 = 9
12
9
12( )5
= 243
log
9243 = 5
2
Common Logarithms
Common Logarithms
Logarithms with a base of 10
Common Logarithms
Logarithms with a base of 10
You will see this one on your calculator
Example 3Solve to the nearest hundredth.
10y = 73
Example 3Solve to the nearest hundredth.
10y = 73
Ok, let’s rewrite this as a logarithm.
Example 3Solve to the nearest hundredth.
10y = 73
Ok, let’s rewrite this as a logarithm.
log 73 = y
Example 3Solve to the nearest hundredth.
10y = 73
Ok, let’s rewrite this as a logarithm.
log 73 = y
Example 3Solve to the nearest hundredth.
10y = 73
Ok, let’s rewrite this as a logarithm.
log 73 = y
Example 3Solve to the nearest hundredth.
10y = 73
Ok, let’s rewrite this as a logarithm.
log 73 = y
y ≈ 1.86
Example 4Solve log t = 2.9 to the nearest tenth.
Example 4Solve log t = 2.9 to the nearest tenth.
Rewrite as a power.
Example 4Solve log t = 2.9 to the nearest tenth.
Rewrite as a power.
102.9 = t
Example 4Solve log t = 2.9 to the nearest tenth.
Rewrite as a power.
102.9 = t
t ≈ 794.3
Properties of Logarithms
Properties of Logarithms
Domain is the set of positive real numbers.
Properties of Logarithms
Domain is the set of positive real numbers.
Range is the set of all real numbers.
Properties of Logarithms
Domain is the set of positive real numbers.
Range is the set of all real numbers.
(1, 0) will be on the graph; logb1 = 0.
Properties of Logarithms
Domain is the set of positive real numbers.
Range is the set of all real numbers.
(1, 0) will be on the graph; logb1 = 0.
The function is strictly increasing.
Properties of Logarithms
Domain is the set of positive real numbers.
Range is the set of all real numbers.
(1, 0) will be on the graph; logb1 = 0.
The function is strictly increasing.
As x increases, y has no bound.
Properties of Logarithms
Properties of Logarithms
As x gets smaller and approaches 0, the values of the function are negative with larger absolute values.
That means when x is between 0 and 1, the exponent will be negative.
Properties of Logarithms
As x gets smaller and approaches 0, the values of the function are negative with larger absolute values.
That means when x is between 0 and 1, the exponent will be negative.
The y-axis is an asymptote.
Homework
Homework
p. 387 #1 - 26
