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DeSmet - Math 152
Blitzer 5E
∫ 9.3 - Logarithmic Functions
1. Logarithmic Functions:
(a) Can we solve the equation 2 x = 10 ?
Above, x is the number you raise 2 by to get 10!
Can we solve 3x = 12 ? x is the number I raise 3 by to get 12.
This is tedious! Since those numbers exist, and are important, we define:
“The number I raise 2 by to get 10 is defined to be log 2 10 .”
“The number I raise 3 by to get 12 is defined to be ______________ .”
(b) Definition: For x > 0 and b > 0 , b ≠ 1 :
b y = x is equivalent to y = log b x
(c) Example 1: Since 2 3 = 8 , we would also write _________________________ .
2. Converting between logarithmic form and exponential form:
Example 2: Write each equation in its equivalent exponential form:
(a)
3 = log b 27
(b )
(c)
log 5 125 = y
7 = log 4 x
Example 3: Write each equation in its equivalent logarithmic form:
(a)
27 = x
Section 9.3
(b )
x3 =
1
2
(c)
e y = 23
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DeSmet - Math 152
Blitzer 5E
3. Example 4: Evaluate each logarithm by inspection.
(a)
log10 (100 )
(b )
log16 ( 4 )
(c)
log 8 ( 8 )
(d )
log 3 ( 27 )
4. Basic Logarithmic Properties:
(a) log b b = ____
(b) log b 1 = ____
5. Logarithm as an Inverse Function:
Let f ( x ) = b x . Find f −1 ( x ) .
Use the above definitions and the fact that f ( f −1 ( x )) = _____ and f −1 ( f ( x )) = _____ to find the
Inverse Properties of Logarithms
6. Example 5: Evaluate each of the following:
(a)
log 9 (1)
Section 9.3
(b )
log16 (16 2 )
(c)
2 log2 12
(d )
log10 ( 0 )
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DeSmet - Math 152
Blitzer 5E
7. Graphs of Logarithmic Functions:
Let f ( x ) = 2 x .
(a) Find f
−1
7
6
( x) .
5
4
3
2
(b) Graph f ( x ) = 2 and f
x
−1
( x ) = ____________
the same axis.
on
1
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
-1
-2
(c) Observations:
-3
-4
y = log 2 (1) = _______
-5
-6
-7
log 2 ( 2 ) = _______
log 2 ( 0 ) = ________
Domain of y = 2 x : ________________
Domain of y = log 2 ( x ) : ______________
Range of y = 2 x : ________________
Range of y = log 2 ( x ) : ______________
In general:
Graph y = b x and y = log b ( x ) on the same axis. We will just consider b > 1 .
Domain of f ( x ) = log b ( x ) : ______________
Range of f ( x ) = log b ( x ) : ______________
Asymptote: _________________
x-intercept: ________________
y-intercept: ________________
Important points:
Section 9.3
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DeSmet - Math 152
Blitzer 5E
8. The Domain of Logarithmic Function:
Consider the function f ( x ) = log b ( g ( x )) . What must be true about g ( x ) to keep f ( x ) happy?
Example 6: Find the domain of f ( x ) = log 2 ( 4 − 3x ) .
9. Common Logarithms:
Base 10 logarithms are called the common logarithms.
In the spirit of laziness, when using common logarithms (logs base 10), we omit the 10, that is
log10 ( x ) is written log ( x ) . (Similar to the square root).
10. Natural Logarithms:
Base e logarithms are called Natural Logarithms.
We write log e x as ln e , that is log e x = ln x .
Both of the above logarithms are very important. So important in fact that they each get their
own button on your calculator!
We will see the importance of logarithms in section 9.5 and 9.6.
11. Example 7: Evaluate each of the following without a calculator.
(a)
⎛ 1⎞
log ⎜ ⎟
⎝ 10 ⎠
(b )
ln
1
e2
(c)
eln
3
2x
(d )
10 log 46
12. (If time) Example 7: Evaluate each of the following using a calculator.
(a)
log 35.8
(b )
ln 4.79
(c)
ln ( −3.5 )
13. Example 8:
As the population of a city increases, the pace of life also increases. The formula
W = 0.35 ln P + 2.74 models average walking speed, W, in feet per second, for a resident of a city
whose population is P thousand. Find the average walking speed in Jackson, Mississippi with a
population of 197 thousand.
Section 9.3
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