Download 171S5.4 Properties of Logarithmic Functions

171S5.4 Properties of Logarithmic Functions
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
Logarithms of Products
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.)
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5.4 Properties of Logarithmic Functions
• Convert from logarithms of products, powers, and quotients to expressions in terms of individual logarithms, and conversely.
• Simplify expressions of the type logaax and .
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.)
Example
Express as a single logarithm:
Solution:
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171S5.4 Properties of Logarithmic Functions
Examples
Express as a product.
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Logarithms of Quotients
The Quotient Rule
For any positive numbers M and N, and any logarithmic base a,
(The logarithm of a quotient is the logarithm of the numerator minus the logarithm of the denominator.)
Example
Express as a difference of logarithms:
Applying the Properties ­ Examples
Express each of the following in terms of sums and differences of logarithms.
Solution:
Example
Express as a single logarithm:
Solution:
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171S5.4 Properties of Logarithmic Functions
Example (continued)
Solution:
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Example (continued)
Solution:
Example
Examples
Given that loga 2 ≈ 0.301 and loga 3 ≈ 0.477, find each of the following, if possible.
Express as a single logarithm:
Solutio
Solution:
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171S5.4 Properties of Logarithmic Functions
Examples (continued)
loga 2 ≈ 0.301 and loga 3 ≈ 0.477
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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.
Cannot be found using these properties and the given information.
(The logarithm, base a, of a to a power is the power.)
Examples
Expressions of the Type
Simplify.
a) loga a8
b) ln e­t c) log 103k
A Base to a Logarithmic Power
For any base a and any positive real number x,
Solution:
a. loga a8
(The number a raised to the power loga x is x.)
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171S5.4 Properties of Logarithmic Functions
Examples
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433/2. Express as the sum of logarithms: log2 (8 . 64)
Simplify.
Solution:
433/14. Express as a product: logb Q ­8
433/18. Express as a difference of logarithms: loga (76 / 13)
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171S5.4 Properties of Logarithmic Functions
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433/32. Express in terms of sums and differences of logarithms: logc ∛(y3 z / x4)
433/24. Express in terms of sums and differences of logarithms: loga x3 y2 z
433/38. Express as a single logarithm and, if possible, simplify:
ln 54 ­ ln 6
433/42. Express as a single logarithm and, if possible, simplify:
(2/5) loga x ­ (1/3) loga y
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171S5.4 Properties of Logarithmic Functions
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433/56. 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) 433/50. Express as a single logarithm and, if possible, simplify:
(2 / 3) [ln (x2 ­ 9) ­ ln (x + 3)] + ln (x + y)
433/68. Simplify: logq q(√ 3) 433/72. Simplify: eln x3 7