Stewart Precalc 6e Section 4.4: Laws of Logarithms

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Stewart Precalc 6e Section 4.4: Laws of Logarithms 0/80
1 Concept - Product Law · Level 1
Fill in the blanks. The logarithm of a product of two numbers is the same as the ___ of the logarithms of these numbers. So \(\log_5(25 \cdot 125) = \) ___ \(+\) ___.
2 Concept - Quotient Law · Level 1
Fill in the blanks. The logarithm of a quotient of two numbers is the same as the ___ of the logarithms of these numbers. So \(\log_5\left(\dfrac{25}{125}\right) = \) ___ \(-\) ___.
3 Concept - Power Law · Level 1
Fill in the blanks. The logarithm of a number raised to a power is the same as the power ___ the logarithm of the number. So \(\log_5(25^{10}) = \) ___ \(\cdot\) ___.
4 Concept - Expanding and Combining · Level 1
(a) We can expand \(\log\left(\dfrac{x^2 y}{z}\right)\) to get ___. (b) We can combine \(2 \log x + \log y - \log z\) to get ___.
5 Concept - Change of Base · Level 1
Fill in the blanks. Most calculators can find logarithms with base ___ and base ___. To find logarithms with different bases, we use the ___ Formula. To find \(\log_7 12\), we write \(\log_7 12 = \dfrac{\log ?}{\log ?}\) which approximately equals ___.
6 Concept - Change of Base · Level 1
True or false? We get the same answer if we do the calculation in Exercise 5 using \(\ln\) in place of \(\log\).
7 Skill - Evaluate · Level 1
Evaluate the expression. \(\log_3 \sqrt{27}\)
8 Skill - Evaluate · Level 1
Evaluate the expression. \(\log_2 160 - \log_2 5\)
9 Skill - Evaluate · Level 1
Evaluate the expression. \(\log 4 + \log 25\)
10 Skill - Evaluate · Level 2
Evaluate the expression. \(\log \dfrac{1}{\sqrt{1000}}\)
11 Skill - Evaluate · Level 1
Evaluate the expression. \(\log_4 192 - \log_4 3\)
12 Skill - Evaluate · Level 1
Evaluate the expression. \(\log_{12} 9 + \log_{12} 16\)
13 Skill - Evaluate · Level 2
Evaluate the expression. \(\log_2 6 - \log_2 15 + \log_2 20\)
14 Skill - Evaluate · Level 2
Evaluate the expression. \(\log_3 100 - \log_3 18 - \log_3 50\)
15 Skill - Evaluate · Level 2
Evaluate the expression. \(\log_4 16^{100}\)
16 Skill - Evaluate · Level 2
Evaluate the expression. \(\log_2 8^{33}\)
17 Skill - Evaluate · Level 2
Evaluate the expression. \(\log(\log 10^{10000})\)
18 Skill - Evaluate · Level 3
Evaluate the expression. \(\ln(\ln e^{e^{200}})\)
19 Skill - Expand · Level 1
Use the Laws of Logarithms to expand the expression. \(\log_2(2 x)\)
20 Skill - Expand · Level 1
Use the Laws of Logarithms to expand the expression. \(\log_3(5 y)\)
21 Skill - Expand · Level 1
Use the Laws of Logarithms to expand the expression. \(\log_2(x(x - 1))\)
22 Skill - Expand · Level 1
Use the Laws of Logarithms to expand the expression. \(\log_5 \dfrac{x}{2}\)
23 Skill - Expand · Level 1
Use the Laws of Logarithms to expand the expression. \(\log 6^{10}\)
24 Skill - Expand · Level 1
Use the Laws of Logarithms to expand the expression. \(\ln \sqrt{z}\)
25 Skill - Expand · Level 2
Use the Laws of Logarithms to expand the expression. \(\log_2(A B^2)\)
26 Skill - Expand · Level 2
Use the Laws of Logarithms to expand the expression. \(\log_6 \sqrt[4]{17}\)
27 Skill - Expand · Level 2
Use the Laws of Logarithms to expand the expression. \(\log_3(x \sqrt{y})\)
28 Skill - Expand · Level 2
Use the Laws of Logarithms to expand the expression. \(\log_2(x y)^{10}\)
29 Skill - Expand · Level 2
Use the Laws of Logarithms to expand the expression. \(\log_5 \sqrt[3]{x^2 + 1}\)
30 Skill - Expand · Level 2
Use the Laws of Logarithms to expand the expression. \(\log_a\left(\dfrac{x^2}{y z^3}\right)\)
31 Skill - Expand · Level 2
Use the Laws of Logarithms to expand the expression. \(\ln \sqrt{a b}\)
32 Skill - Expand · Level 2
Use the Laws of Logarithms to expand the expression. \(\ln \sqrt[3]{3 r^2 s}\)
33 Skill - Expand · Level 2
Use the Laws of Logarithms to expand the expression. \(\log\left(\dfrac{x^3 y^4}{z^6}\right)\)
34 Skill - Expand · Level 2
Use the Laws of Logarithms to expand the expression. \(\log\left(\dfrac{a^2}{b^4 \sqrt{c}}\right)\)
35 Skill - Expand · Level 3
Use the Laws of Logarithms to expand the expression. \(\log_2(\dfrac{x(x^2 + 1)}{\sqrt{x^2 - 1}})\)
36 Skill - Expand · Level 3
Use the Laws of Logarithms to expand the expression. \(\log_5 \sqrt{\dfrac{x - 1}{x + 1}}\)
37 Skill - Expand · Level 3
Use the Laws of Logarithms to expand the expression. \(\ln\left(x \sqrt{\dfrac{y}{z}}\right)\)
38 Skill - Expand · Level 3
Use the Laws of Logarithms to expand the expression. \(\ln \dfrac{3 x^2}{(x + 1)^{10}}\)
39 Skill - Expand · Level 2
Use the Laws of Logarithms to expand the expression. \(\log \sqrt[4]{x^2 + y^2}\)
40 Skill - Expand · Level 3
Use the Laws of Logarithms to expand the expression. \(\log\left(\dfrac{x}{\sqrt[3]{1 - x}}\right)\)
41 Skill - Expand · Level 4
Use the Laws of Logarithms to expand the expression. \(\log \sqrt{\dfrac{x^2 + 4}{(x^2 + 1)(x^3 - 7)^2}}\)
42 Skill - Expand · Level 4
Use the Laws of Logarithms to expand the expression. \(\log \sqrt{x \sqrt{y \sqrt{z}}}\)
43 Skill - Expand · Level 3
Use the Laws of Logarithms to expand the expression. \(\ln\left(\dfrac{x^3 \sqrt{x - 1}}{3 x + 4}\right)\)
44 Skill - Expand · Level 3
Use the Laws of Logarithms to expand the expression. \(\log(\dfrac{10^x}{x(x^2 + 1)(x^4 + 2)})\)
45 Skill - Combine · Level 2
Use the Laws of Logarithms to combine the expression. \(\log_3 5 + 5 \log_3 2\)
46 Skill - Combine · Level 2
Use the Laws of Logarithms to combine the expression. \(\log 12 + \dfrac{1}{2} \log 7 - \log 2\)
47 Skill - Combine · Level 2
Use the Laws of Logarithms to combine the expression. \(\log_2 A + \log_2 B - 2 \log_2 C\)
48 Skill - Combine · Level 2
Use the Laws of Logarithms to combine the expression. \(\log_5(x^2 - 1) - \log_5(x - 1)\)
49 Skill - Combine · Level 3
Use the Laws of Logarithms to combine the expression. \(4 \log x - \dfrac{1}{3} \log(x^2 + 1) + 2 \log(x - 1)\)
50 Skill - Combine · Level 2
Use the Laws of Logarithms to combine the expression. \(\ln(a + b) + \ln(a - b) - 2 \ln c\)
51 Skill - Combine · Level 2
Use the Laws of Logarithms to combine the expression. \(\ln 5 + 2 \ln x + 3 \ln(x^2 + 5)\)
52 Skill - Combine · Level 3
Use the Laws of Logarithms to combine the expression. \(2(\log_5 x + 2 \log_5 y - 3 \log_5 z)\)
53 Skill - Combine · Level 2
Use the Laws of Logarithms to combine the expression. \(\log_a b + c \log_a d - r \log_a s\)
54 Skill - Change of Base · Level 2
Use the Change of Base Formula and a calculator to evaluate the logarithm, rounded to six decimal places. Use either natural or common logarithms. \(\log_2 5\)
55 Skill - Change of Base · Level 2
Use the Change of Base Formula and a calculator to evaluate the logarithm, rounded to six decimal places. \(\log_5 2\)
56 Skill - Change of Base · Level 2
Use the Change of Base Formula and a calculator to evaluate the logarithm, rounded to six decimal places. \(\log_3 16\)
57 Skill - Change of Base · Level 2
Use the Change of Base Formula and a calculator to evaluate the logarithm, rounded to six decimal places. \(\log_6 92\)
58 Skill - Change of Base · Level 2
Use the Change of Base Formula and a calculator to evaluate the logarithm, rounded to six decimal places. \(\log_7 2.61\)
59 Skill - Change of Base · Level 2
Use the Change of Base Formula and a calculator to evaluate the logarithm, rounded to six decimal places. \(\log_6 532\)
60 Skill - Change of Base · Level 2
Use the Change of Base Formula and a calculator to evaluate the logarithm, rounded to six decimal places. \(\log_4 125\)
61 Skill - Change of Base · Level 2
Use the Change of Base Formula and a calculator to evaluate the logarithm, rounded to six decimal places. \(\log_{12} 2.5\)
62 Skill - Change of Base (Show) · Level 2
Use the Change of Base Formula to show that \(\log_3 x = \dfrac{\ln x}{\ln 3}\). Then use this fact to draw the graph of the function \(f(x) = \log_3 x\).
63 Skill - Graphing Family of Logs · Level 3
Draw graphs of the family of functions \(y = \log_a x\) for \(a = 2, e, 5,\) and \(10\) on the same screen, using the viewing rectangle \([0, 5]\) by \([-3, 3]\). How are these graphs related?
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64 Skill - Change of Base (Show) · Level 2
Use the Change of Base Formula to show that \(\log e = \dfrac{1}{\ln 10}\).
65 Skill - Simplify Product · Level 3
Simplify: \((\log_2 5)(\log_5 7)\).
66 Skill - Logarithm Identity · Level 3
Show that \(-\ln(x - \sqrt{x^2 - 1}) = \ln(x + \sqrt{x^2 - 1})\).
67 Application - Forgetting · Level 3
Use the Law of Forgetting (Example
5) to estimate a student's score on a biology test two years after he got a score of \(80\) on a test covering the same material. Assume that \(c = 0.3\) and \(t\) is measured in months.
68 Application - Wealth Distribution · Level 3
Vilfredo Pareto (1848–1923) observed that most of the wealth of a country is owned by a few members of the population. Pareto's Principle is \(\log P = \log c - k \log W\) where \(W\) is the wealth level (how much money a person has) and \(P\) is the number of people in the population having that much money. (a) Solve the equation for \(P\). (b) Assume that \(k = 2.1\), \(c = 8000\), and \(W\) is measured in millions of dollars. Use part (a) to find the number of people who have \$2 million or more. How many people have \$10 million or more?
69 Application - Biodiversity · Level 3
Some biologists model the number of species \(S\) in a fixed area \(A\) (such as an island) by the species-area relationship \(\log S = \log c + k \log A\) where \(c\) and \(k\) are positive constants that depend on the type of species and habitat. (a) Solve the equation for \(S\). (b) Use part (a) to show that if \(k = 3\), then doubling the area increases the number of species eightfold.
70 Application - Magnitude of Stars · Level 3
The magnitude \(M\) of a star is a measure of how bright a star appears to the human eye. It is defined by \(M = -2.5 \log\left(\dfrac{B}{B_0}\right)\) where \(B\) is the actual brightness of the star and \(B_0\) is a constant. (a) Expand the right-hand side of the equation. (b) Use part (a) to show that the brighter a star, the less its magnitude. (c) Betelgeuse is about \(100\) times brighter than Albiero. Use part (a) to show that Betelgeuse is \(5\) magnitudes less bright than Albiero.
71 Discovery - True or False · Level 2
Discuss each equation and determine whether it is true for all possible values of the variables. (Ignore values of the variables for which any term is undefined.) (a) \(\log\left(\dfrac{x}{y}\right) = \dfrac{\log x}{\log y}\) (b) \(\log_2(x - y) = \log_2 x - \log_2 y\) (c) \(\log_5\left(\dfrac{a}{b^2}\right) = \log_5 a - 2 \log_5 b\) (d) \(\log 2^z = z \log 2\) (e) \((\log P)(\log Q) = \log P + \log Q\) (f) \(\dfrac{\log a}{\log b} = \log a - \log b\) (g) \((\log_2 7)^x = x \log_2 7\) (h) \(\log_a a^a = a\) (i) \(\log(x - y) = \dfrac{\log x}{\log y}\) (j) \(-\ln\left(\dfrac{1}{A}\right) = \ln A\)
72 Discovery/Discussion/Writing - Find the Error · Level 4
*Find the Error.* What is wrong with the following argument? \(\log 0.1 < 2 \log 0.1 = \log(0.1)^2 = \log 0.01\) Therefore \(\log 0.1 < \log 0.01\), which gives \(0.1 < 0.01\).
73 Discovery/Discussion/Writing - Graph Transformations · Level 4
*Shifting, Shrinking, and Stretching Graphs of Functions.* Let \(f(x) = x^2\). Show that \(f(2 x) = 4 f(x)\), and explain how this shows that shrinking the graph of \(f\) horizontally has the same effect as stretching it vertically. Then use the identities \(e^{2 + x} = e^2 e^x\) and \(\ln(2 x) = \ln 2 + \ln x\) to show that for \(g(x) = e^x\) a horizontal shift is the same as a vertical stretch and for \(h(x) = \ln x\) a horizontal shrinking is the same as a vertical shift.
74 Example - Using Laws of Logarithms to Evaluate Expressions · Level 2
Evaluate each expression. (a) \(\log_4 2 + \log_4 32\) (b) \(\log_2 80 - \log_2 5\) (c) \(-\dfrac{1}{3} \log 8\)
75 Example - Expanding Logarithmic Expressions · Level 2
Use the Laws of Logarithms to expand each expression. (a) \(\log_2(6 x)\) (b) \(\log_5(x^3 y^6)\) (c) \(\ln\left(\dfrac{a b}{\sqrt[3]{c}}\right)\)
76 Example - Combining Logarithmic Expressions · Level 2
Combine \(3 \log x + \dfrac{1}{2} \log(x + 1)\) into a single logarithm.
77 Example - Combining Logarithmic Expressions · Level 3
Combine \(3 \ln s + \dfrac{1}{2} \ln t - 4 \ln(t^2 + 1)\) into a single logarithm.
78 Example - The Law of Forgetting (Application) · Level 3
If a task is learned at a performance level \(P_0\), then after a time interval \(t\) the performance level \(P\) satisfies \(\log P = \log P_0 - c \log(t + 1)\) where \(c\) is a constant that depends on the type of task and \(t\) is measured in months. (a) Solve for \(P\). (b) If your score on a history test is \(90\), what score would you expect to get on a similar test after two months? After a year? (Assume that \(c = 0.2\).)
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79 Example - Evaluating Logarithms with the Change of Base Formula · Level 2
Use the Change of Base Formula and common or natural logarithms to evaluate each logarithm, correct to five decimal places. (a) \(\log_8 5\) (b) \(\log_9 20\)
80 Example - Graphing Logarithms with Change of Base · Level 2
Use a graphing calculator to graph \(f(x) = \log_6 x\).

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