Download 171S5.4q Properties of Logarithmic Functions

171S5.4q Properties of Logarithmic Functions
April 16, 2013
Logarithms of Products
MAT 171 Precalculus Algebra
Dr. Claude Moore
Cape Fear Community College
CHAPTER 5: Exponential and Logarithmic Functions
5.1 Inverse Functions
5.2 Exponential Functions and Graphs
5.3 Logarithmic Functions and Graphs
5.4 Properties of Logarithmic Functions
5.5 Solving Exponential and Logarithmic Equations 5.6 Applications and Models: Growth and Decay; and Compound Interest
5.4 Properties of Logarithmic Functions
• Convert from logarithms of products, powers, and quotients to expressions in terms of individual logarithms, and conversely.
The Product Rule
For any positive numbers M and N and any logarithmic base a, loga MN = loga M + loga N.
(The logarithm of a product is the sum of the logarithms of the factors.)
Example
Express as a single logarithm:
Solution:
Logarithms of Powers
The Power Rule For any positive number M, any logarithmic base a, and any real number p,
(The logarithm of a power of M is the exponent times the logarithm of M.)
• Simplify expressions of the type logaax and .
Nov 15­1:50 PM
Nov 15­1:50 PM
Logarithms of Quotients
Examples
The Quotient Rule
For any positive numbers M and N, and any logarithmic base a,
Express as a product.
(The logarithm of a quotient is the logarithm of the numerator minus the logarithm of the denominator.)
Example
Express as a difference of logarithms:
Solution:
Example
Express as a single logarithm:
Solution:
Nov 15­1:50 PM
Applying the Properties ­ Examples
Express each of the following in terms of sums and differences of logarithms.
Nov 15­1:50 PM
Example (continued)
Solution:
Solution:
Example
Express as a single logarithm:
Solution:
Nov 15­1:50 PM
Nov 15­1:50 PM
1
171S5.4q Properties of Logarithmic Functions
Examples
Given that loga 2 ≈ 0.301 and loga 3 ≈ 0.477, find each of the following, if possible.
April 16, 2013
Expressions of the Type loga ax
The Logarithm of a Base to a Power
For any base a and any real number x,
loga ax = x.
Solution:
(The logarithm, base a, of a to a power is the power.)
Examples
Simplify.
a) loga a8 b) ln e­t c) log 103k
Solution:
Cannot be found using these properties and the given information.
Nov 15­1:50 PM
Expressions of the Type
a. loga a8
Nov 15­1:50 PM
441/2. Express as the sum of logarithms: log2 (8 . 64)
A Base to a Logarithmic Power
For any base a and any positive real number x,
(The number a raised to the power loga x is x.)
441/4. Express as the sum of logarithms: log4 (64 . 4)
Examples
Simplify.
441/6. Express as the sum of logarithms: log 0.2x
Solution:
441/8. Express as the sum of logarithms:
ln ab
Nov 15­1:50 PM
441/10. Express as a product: loga x 4
Nov 15­8:13 PM
441/18. Express as a difference of logarithms: loga (76 / 13)
441/12. Express as a product: ln y 5
441/20. Express as a difference of logarithms: ln (a / b)
441/14. Express as a product: logb Q ­8
441/21. Express as a difference of logarithms:
ln (r / s)
441/16. Express as a product: ln √a
Nov 15­8:13 PM
441/22. Express as a difference of logarithms:
logb (3 / w)
Nov 15­8:13 PM
2
171S5.4q Properties of Logarithmic Functions
441/24. Express in terms of sums and differences of logarithms: 441/26. Express in terms of sums and differences of logarithms:
Nov 15­8:13 PM
441/38. Express as a single logarithm and, if possible, simplify:
ln 54 ­ ln 6
441/42. Express as a single logarithm and, if possible, simplify:
(2/5) loga x ­ (1/3) loga y
April 16, 2013
441/32. Express in terms of sums and differences of logarithms: 441/34. Express in terms of sums and differences of logarithms: Nov 15­8:13 PM
442/48. Express as a single logarithm and, if possible, simplify:
442/50. Express as a single logarithm and, if possible, simplify:
(2 / 3) [ln (x2 ­ 9) ­ ln (x + 3)] + ln (x + y)
Nov 15­8:13 PM
Nov 15­8:13 PM
442/54. Given that loga 2 ≈ 0.301, loga 7 ≈ 442/57. Given that loga 2 ≈ 0.301, loga 7 ≈ 0.845, and loga 11 ≈ 1.041, find each of the following, if possible. Round to nearest thousandth:
loga 14 0.845, and loga 11 ≈ 1.041, find each of the following, if possible. Round to nearest thousandth:
442/56. Given that loga 2 ≈ 0.301, loga 7 ≈ 0.845, and loga 11 ≈ 1.041, find each of the following, if possible. Round to nearest thousandth:
loga (1 / 7) Nov 15­8:13 PM
442/58. Given that loga 2 ≈ 0.301, loga 7 ≈ 0.845, and loga 11 ≈ 1.041, find each of the following, if possible. Round to nearest thousandth:
loga 9 Nov 15­8:13 PM
3
171S5.4q Properties of Logarithmic Functions
April 16, 2013
442/68. Simplify: logq q(√ 3) 442/70. Simplify: Nov 15­8:13 PM
Apr 16­8:41 AM
442/72. Simplify: 442/74. Simplify: log 10 ­k Nov 15­8:13 PM
Apr 16­4:12 PM
4