1. How does the power rule for logarithms help when solving logarithms with the form logb(nx)?
2. What does the change-of-base formula do? Why is it useful when using a calculator?
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)
4. ln(3ab⋅5c)
5. logb(1713)
6. log4(wzx)
7. ln(4k1)
8. log2(yx)
For the following exercises, condense to a single logarithm if possible.
9. ln(7)+ln(x)+ln(y)
10. log3(2)+log3(a)+log3(11)+log3(b)
11. logb(28)−logb(7)
12. ln(a)−ln(d)−ln(c)
13. −logb(71)
14. 31ln(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(z19x15y13)
16. ln(b−4c5a−2)
17. log(x3y−4)
18. ln(y1−yy)
19. log(x2y33x2y5)
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)
22. 2log(x)+3log(x+1)
23. log(x)−21log(y)+3log(z)
24. 4log7(c)+3log7(a)+3log7(b)
For the following exercises, rewrite each expression as an equivalent ratio of logs using the indicated base.
25. log7(15) to base e
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)
28. log6(55)
29. log11(116)
For the following exercises, use properties of logarithms to evaluate without using a calculator.
30. log3(91)−3log3(3)
31. 6log8(2)+3log8(4)log8(64)
32. 2log9(3)−4log9(3)+log9(7291)
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)
34. log8(65)
35. log6(5.38)
36. log4(215)
37. log21(4.7)
38. Use the product rule for logarithms to find all x values such that log12(2x+6)+log12(x+2)=2. Show the steps for solving.
39. Use the quotient rule for logarithms to find all x values such that log6(x+2)−log6(x−3)=1. Show the steps for solving.
40. Can the power property of logarithms be derived from the power property of exponents using the equation bx=m? If not, explain why. If so, show the derivation.
41. Prove that logb(n)=logn(b)1 for any positive integers b > 1 and n > 1.
42. Does log81(2401)=log3(7)? Verify the claim algebraically.
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Precalculus.Provided by: OpenStaxAuthored by: Jay Abramson, et al..Located at: https://openstax.org/books/precalculus/pages/1-introduction-to-functions.License: CC BY: Attribution. License terms: Download For Free at : http://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175..