5.6E: Exercises
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- Jan 20, 2020
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4.5: Logarithmic Properties
Verbal
1) How does the power rule for logarithms help when solving logarithms with the form logb(n√x)
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Any root expression can be rewritten as an expression with a rational exponent so that the power rule can be applied, making the logarithm easier to calculate. Thus, logb(x1n)=1nlogb(x).
2) What does the change-of-base formula do? Why is it useful when using a calculator?
Algebraic
For the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.
3) logb(7x⋅2y)
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logb(2)+logb(7)+logb(x)+logb(y)
4) ln(3ab⋅5c)
5) logb(1317)
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logb(13)−logb(17)
6) log4(xzw)
7) ln(14k)
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−kln(4)
8) log2(yx)
For the following exercises, condense to a single logarithm if possible.
9) ln(7)+ln(x)+ln(y)
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ln(7xy)
10) log3(2)+log3(a)+log3(11)+log3(b)
11) logb(28)−logb(7)
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logb(4)
12) ln(a)−ln(d)−ln(c)
13) −logb(17)
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logb(7)
14) 13ln(8)
For the following exercises, use the properties of logarithms to expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.
15) log(x15y13z19)
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15log(x)+13log(y)−19log(z)
16) ln(a−2b−4c5)
17) log(√x3y−4)
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32log(x)−2log(y)
18) ln(y√y1−y)
19) log(x2y33√x2y5)
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83log(x)+143log(y)
For the following exercises, condense each expression to a single logarithm using the properties of logarithms.
20) log(2x4)+log(3x5)
21) ln(6x9)−ln(3x2)
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ln(2x7)
22) 2log(x)+3log(x+1)
23) log(x)−12log(y)+3log(z)
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log(xz3√y)
24) 4log7(c)+log7(a)3+log7(b)3
For the following exercises, rewrite each expression as an equivalent ratio of logs using the indicated base.
25) log7(15) to base e
- Answer
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log7(15)=ln(15)ln(7)
26) log14(55.875) to base 10
For the following exercises, suppose log5(6)=a and log5(11)=b Use the change-of-base formula along with properties of logarithms to rewrite each expression in terms of a and b Show the steps for solving.
27) log11(5)
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log11(5)=log5(5)log5(11)=1b
28) log6(55)
29) log11(611)
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log11(611)=log11(611)log5(11)=log5(6)−log5(11)log5(11)=a−bb=ab−1
Numeric
For the following exercises, use properties of logarithms to evaluate without using a calculator.
30) log3(19)−3log3(3)
31) 6log8(2)+log8(64)3log8(4)
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3
32) 2log9(3)−4log9(3)+log9(1729)
For the following exercises, use the change-of-base formula to evaluate each expression as a quotient of natural logs. Use a calculator to approximate each to five decimal places.
33) log3(22)
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2.81359
34) log8(65)
35) log6(5.38)
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0.93913
36) log4(152)
37) log12(4.7)
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−2.23266
Extensions
38) Use the product rule for logarithms to find all x values such that log12(2x+6)+log12(x+2)=2Show the steps for solving.
39) Use the quotient rule for logarithms to find all x values such that log6(x+2)−log6(x−3)=1Show the steps for solving.
- Answer
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Rewriting as an exponential equation and solving for x:
61=x+2x−30=x+2x−3−60=x+2x−3−6(x−3)(x−3)0=x+2−6x+18x−30=x−4x−3x=4
Checking, we find that log6(4+2)−log6(4−3)=log6(6)−log6(1) is defined, so x=4
40) Can the power property of logarithms be derived from the power property of exponents using the equation bx=mIf not, explain why. If so, show the derivation.
41) Prove that logb(n)=1logb(n) for any positive integers b>1 and n>1
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Let b and n be positive integers greater than 1Then, by the change-of-base formula, logb(n)=logn(n)logn(b)=1logn(b)
42) Does log81(2401)=log3(7)Verify the claim algebraically.