4.3 Rules of Logarithms. Definition of a Logarithmic Function For x > 0 and b > 0, b 1, y = log b x...
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Transcript of 4.3 Rules of Logarithms. Definition of a Logarithmic Function For x > 0 and b > 0, b 1, y = log b x...
![Page 1: 4.3 Rules of Logarithms. Definition of a Logarithmic Function For x > 0 and b > 0, b 1, y = log b x is equivalent to b y = x. The function f (x) = log.](https://reader036.fdocuments.net/reader036/viewer/2022082512/5514462e5503466d1a8b5a88/html5/thumbnails/1.jpg)
4.3 Rules of Logarithms
![Page 2: 4.3 Rules of Logarithms. Definition of a Logarithmic Function For x > 0 and b > 0, b 1, y = log b x is equivalent to b y = x. The function f (x) = log.](https://reader036.fdocuments.net/reader036/viewer/2022082512/5514462e5503466d1a8b5a88/html5/thumbnails/2.jpg)
Definition of a Logarithmic Function
• For x > 0 and b > 0, b 1,• y = logb x is equivalent to by = x.• The function f (x) = logb x is the
logarithmic function with base b.
![Page 3: 4.3 Rules of Logarithms. Definition of a Logarithmic Function For x > 0 and b > 0, b 1, y = log b x is equivalent to b y = x. The function f (x) = log.](https://reader036.fdocuments.net/reader036/viewer/2022082512/5514462e5503466d1a8b5a88/html5/thumbnails/3.jpg)
Location of Base and Exponent in Exponential and Logarithmic Forms
Logarithmic form: y = logb x Exponential Form: by = x. Logarithmic form: y = logb x Exponential Form: by = x.
Exponent Exponent
Base Base
![Page 4: 4.3 Rules of Logarithms. Definition of a Logarithmic Function For x > 0 and b > 0, b 1, y = log b x is equivalent to b y = x. The function f (x) = log.](https://reader036.fdocuments.net/reader036/viewer/2022082512/5514462e5503466d1a8b5a88/html5/thumbnails/4.jpg)
Text Example
Write each equation in its equivalent exponential form.a. 2 = log5 x b. 3 = logb 64 c. log3 7 = y
Solution With the fact that y = logb x means by = x,
c. log3 7 = y or y = log3 7 means 3y = 7.
a. 2 = log5 x means 52 = x.
Logarithms are exponents.Logarithms are exponents.
b. 3 = logb 64 means b3 = 64.
Logarithms are exponents.Logarithms are exponents.
![Page 5: 4.3 Rules of Logarithms. Definition of a Logarithmic Function For x > 0 and b > 0, b 1, y = log b x is equivalent to b y = x. The function f (x) = log.](https://reader036.fdocuments.net/reader036/viewer/2022082512/5514462e5503466d1a8b5a88/html5/thumbnails/5.jpg)
Evaluatea. log2 16 b. log3 9 c. log25 5
Solution
log25 5 = 1/2 because 251/2 = 5.25 to what power is 5?c. log25 5
log3 9 = 2 because 32 = 9.3 to what power is 9?b. log3 9
log2 16 = 4 because 24 = 16.2 to what power is 16?a. log2 16
Logarithmic Expression Evaluated
Question Needed for Evaluation
Logarithmic Expression
Text Example
![Page 6: 4.3 Rules of Logarithms. Definition of a Logarithmic Function For x > 0 and b > 0, b 1, y = log b x is equivalent to b y = x. The function f (x) = log.](https://reader036.fdocuments.net/reader036/viewer/2022082512/5514462e5503466d1a8b5a88/html5/thumbnails/6.jpg)
Basic Logarithmic Properties Involving One
• Logb b = 1 because 1 is the exponent to which b must be raised to obtain b. (b1 = b).
• Logb 1 = 0 because 0 is the exponent to which b must be raised to obtain 1. (b0 = 1).
![Page 7: 4.3 Rules of Logarithms. Definition of a Logarithmic Function For x > 0 and b > 0, b 1, y = log b x is equivalent to b y = x. The function f (x) = log.](https://reader036.fdocuments.net/reader036/viewer/2022082512/5514462e5503466d1a8b5a88/html5/thumbnails/7.jpg)
Inverse Properties of Logarithms
For x > 0 and b 1,
logb bx = x The logarithm with base b of b raised to a power equals that power.b logb x = x b raised to the logarithm with base b of a number equals that number.
![Page 8: 4.3 Rules of Logarithms. Definition of a Logarithmic Function For x > 0 and b > 0, b 1, y = log b x is equivalent to b y = x. The function f (x) = log.](https://reader036.fdocuments.net/reader036/viewer/2022082512/5514462e5503466d1a8b5a88/html5/thumbnails/8.jpg)
Graph f (x) = 2x and g(x) = log2 x in the same rectangular coordinate system.
Solution We first set up a table of coordinates for f (x) = 2x. Reversing these coordinates gives the coordinates for the inverse function, g(x) = log2 x.
4
2
8211/21/4f (x) = 2x
310-1-2x
2
4
310-1-2g(x) = log2 x
8211/21/4x
Reverse coordinates.
Text Example
![Page 9: 4.3 Rules of Logarithms. Definition of a Logarithmic Function For x > 0 and b > 0, b 1, y = log b x is equivalent to b y = x. The function f (x) = log.](https://reader036.fdocuments.net/reader036/viewer/2022082512/5514462e5503466d1a8b5a88/html5/thumbnails/9.jpg)
Graph f (x) = 2x and g(x) = log2 x in the same rectangular coordinate system.
Solution
We now plot the ordered pairs in both tables, connecting them with smooth curves. The graph of the inverse can also be drawn by reflecting the graph of f (x) = 2x over the line y = x.
-2 -1
6
2 3 4 5
5
4
3
2
-1-2
6
f (x) = 2x
f (x) = log2 x
y = x
Text Example cont.
![Page 10: 4.3 Rules of Logarithms. Definition of a Logarithmic Function For x > 0 and b > 0, b 1, y = log b x is equivalent to b y = x. The function f (x) = log.](https://reader036.fdocuments.net/reader036/viewer/2022082512/5514462e5503466d1a8b5a88/html5/thumbnails/10.jpg)
Characteristics of the Graphs of Logarithmic Functions of the Form f(x) = logbx
• The x-intercept is 1. There is no y-intercept.• The y-axis is a vertical asymptote.• If b > 1, the function is increasing. If 0 < b < 1,
the function is decreasing.• The graph is smooth and continuous. It has no
sharp corners or edges.
![Page 11: 4.3 Rules of Logarithms. Definition of a Logarithmic Function For x > 0 and b > 0, b 1, y = log b x is equivalent to b y = x. The function f (x) = log.](https://reader036.fdocuments.net/reader036/viewer/2022082512/5514462e5503466d1a8b5a88/html5/thumbnails/11.jpg)
Properties of Common Logarithms
General Properties Common Logarithms
1. logb 1 = 0 1. log 1 = 0
2. logb b = 1 2. log 10 = 1
3. logb bx = x 3. log 10x = x4. b logb x = x 4. 10 log x = x
![Page 12: 4.3 Rules of Logarithms. Definition of a Logarithmic Function For x > 0 and b > 0, b 1, y = log b x is equivalent to b y = x. The function f (x) = log.](https://reader036.fdocuments.net/reader036/viewer/2022082512/5514462e5503466d1a8b5a88/html5/thumbnails/12.jpg)
Examples of Logarithmic Properties
log b b = 1
log b 1 = 0
log 4 4 = 1
log 8 1 = 0
3 log 3 6 = 6
log 5 5 3 = 3
2 log 2 7 = 7
![Page 13: 4.3 Rules of Logarithms. Definition of a Logarithmic Function For x > 0 and b > 0, b 1, y = log b x is equivalent to b y = x. The function f (x) = log.](https://reader036.fdocuments.net/reader036/viewer/2022082512/5514462e5503466d1a8b5a88/html5/thumbnails/13.jpg)
Properties of Natural Logarithms
General Properties Natural Logarithms
1. logb 1 = 0 1. ln 1 = 0
2. logb b = 1 2. ln e = 1
3. logb bx = x 3. ln ex = x4. b logb x = x 4. e ln x = x
![Page 14: 4.3 Rules of Logarithms. Definition of a Logarithmic Function For x > 0 and b > 0, b 1, y = log b x is equivalent to b y = x. The function f (x) = log.](https://reader036.fdocuments.net/reader036/viewer/2022082512/5514462e5503466d1a8b5a88/html5/thumbnails/14.jpg)
Examples of Natural Logarithmic Properties
log e e = 1
log e 1 = 0
e log e 6 = 6
log e e 3 = 3
![Page 15: 4.3 Rules of Logarithms. Definition of a Logarithmic Function For x > 0 and b > 0, b 1, y = log b x is equivalent to b y = x. The function f (x) = log.](https://reader036.fdocuments.net/reader036/viewer/2022082512/5514462e5503466d1a8b5a88/html5/thumbnails/15.jpg)
4.3 Rules of Logarithms