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Mathematics LibreTexts

5.6E: Exercises

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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(nx)

Answer

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(7x2y)

Answer

logb(2)+logb(7)+logb(x)+logb(y)

4) ln(3ab5c)

5) logb(1317)

Answer

logb(13)logb(17)

6) log4(xzw)

7) ln(14k)

Answer

kln(4)

8) log2(yx)

For the following exercises, condense to a single logarithm if possible.

9) ln(7)+ln(x)+ln(y)

Answer

ln(7xy)

10) log3(2)+log3(a)+log3(11)+log3(b)

11) logb(28)logb(7)

Answer

logb(4)

12) ln(a)ln(d)ln(c)

13) logb(17)

Answer

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)

Answer

15log(x)+13log(y)19log(z)

16) ln(a2b4c5)

17) log(x3y4)

Answer

32log(x)2log(y)

18) ln(yy1y)

19) log(x2y33x2y5)

Answer

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)

Answer

ln(2x7)

22) 2log(x)+3log(x+1)

23) log(x)12log(y)+3log(z)

Answer

log(xz3y)

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

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)

Answer

log11(5)=log5(5)log5(11)=1b

28) log6(55)

29) log11(611)

Answer

log11(611)=log11(611)log5(11)=log5(6)log5(11)log5(11)=abb=ab1

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)

Answer

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)

Answer

2.81359

34) log8(65)

35) log6(5.38)

Answer

0.93913

36) log4(152)

37) log12(4.7)

Answer

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(x3)=1Show the steps for solving.

Answer

Rewriting as an exponential equation and solving for x:

61=x+2x30=x+2x360=x+2x36(x3)(x3)0=x+26x+18x30=x4x3x=4

Checking, we find that log6(4+2)log6(43)=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

Answer

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.


5.6E: Exercises is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.

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