1. Using Properties of Logarithms

3. Example
Use the product rule to expand each logarithmic expression.
𝑙𝑜𝑔 3 9·5
𝑙𝑜𝑔 1000𝑥

4. Solution
Use the product rule to expand each logarithmic expression.
𝑙𝑜𝑔 3 9·5 = 𝑙𝑜𝑔 3 9 · 𝑙𝑜𝑔 3 5
𝑙𝑜𝑔 1000𝑥 =𝑙𝑜𝑔 𝑙𝑜𝑔 𝑥

6. Example
Use the quotient rule to expand each logarithmic expression.
𝑙𝑜𝑔 𝑥
𝑙𝑜𝑔 𝑥 8
𝑙𝑛 𝑒 3 7

7. Solution
Use the quotient rule to expand each logarithmic expression.
𝑙𝑜𝑔 𝑥 = 𝑙𝑜𝑔 − 𝑙𝑜𝑔 5 𝑥
𝑙𝑜𝑔 𝑥 8 =𝑙𝑜𝑔 𝑥 −𝑙𝑜𝑔 8
𝑙𝑛 𝑒 =𝑙𝑛 𝑒 3 −𝑙𝑛 7

9. Example
Use the power rule to expand each logarithmic expression.
𝑙𝑜𝑔
𝑙𝑜𝑔 2 8𝑥 4
𝑙𝑜𝑔 𝑥
𝑙𝑛 6𝑒 5

10. Solution
Use the power rule to expand each logarithmic expression.
𝑙𝑜𝑔 =2 𝑙𝑜𝑔 5 7
𝑙𝑜𝑔 2 8𝑥 4 =4 𝑙𝑜𝑔 2 8𝑥
𝑙𝑜𝑔 𝑥 = 1 2 𝑙𝑜𝑔𝑥
𝑙𝑛 6𝑒 5 =5𝑙𝑛 6𝑒

11. Expanding Logarithmic Expressions

13. Study Tip

14. Example
Use logarithmic properties to expand each expression as much as possible.
𝑙𝑜𝑔 𝑏 𝑥 𝑧
𝑙𝑜𝑔 𝑦 3
𝑙𝑛 10 𝑒
𝑙𝑜𝑔

15. Solutions
Use logarithmic properties to expand each expression as much as possible.
𝑙𝑜𝑔 𝑏 𝑥 𝑧 2 =2 𝑙𝑜𝑔 𝑏 𝑥 𝑙𝑜𝑔 𝑏 𝑧 =2 𝑙𝑜𝑔 𝑏 𝑥+ 𝑙𝑜𝑔 𝑏 𝑧
𝑙𝑜𝑔 𝑦 3 =3 𝑙𝑜𝑔 5 25− 𝑙𝑜𝑔 5 𝑦 =3 2− 𝑙𝑜𝑔 5 𝑦 =6−3 𝑙𝑜𝑔 5 𝑦
𝑙𝑛 10 𝑒 =𝑙𝑛 10 −𝑙𝑛 𝑒
=𝑙𝑛 10 −1
𝑙𝑜𝑔 =𝑙𝑜𝑔 𝑙𝑜𝑔 10
=𝑙𝑜𝑔

16. Condensing Logarithmic Expressions

18. Example
Write as a single logarithm (condense)
𝑙𝑜𝑔 𝑙𝑜𝑔
𝑙𝑜𝑔 6𝑥 −𝑙𝑜𝑔 6
2 𝑙𝑜𝑔 3 9− 𝑙𝑜𝑔 3 27
𝑙𝑛 𝑥−2 +5𝑙𝑛 𝑥

19. Solution Write as a single logarithm (condense)
𝑙𝑜𝑔 𝑙𝑜𝑔 = 𝑙𝑜𝑔
𝑙𝑜𝑔 6𝑥 −𝑙𝑜𝑔 6=𝑙𝑜𝑔 6𝑥 6
=𝑙𝑜𝑔 𝑥
2 𝑙𝑜𝑔 3 9− 𝑙𝑜𝑔 3 27= 𝑙𝑜𝑔
= 𝑙𝑜𝑔 3 3 =1
𝑙𝑛 𝑥−2 +5𝑙𝑛 𝑥=𝑙𝑛 𝑥−2 · 𝑥 5
=𝑙𝑛 𝑥 5 𝑥−2

20. Question Can you simplify this any further?
𝑙𝑜𝑔 𝑙𝑜𝑔 = 𝑙𝑜𝑔

21. Question Can you simplify this any further?
𝑙𝑜𝑔 𝑙𝑜𝑔 = 𝑙𝑜𝑔
How about this:
𝑙𝑜𝑔 𝑙𝑜𝑔 = 𝑙𝑜𝑔 𝑙𝑜𝑔 = 𝑙𝑜𝑔 𝑙𝑜𝑔 = = 7 2

22. The Change-of-Base Property

23. The Change-of-Base Property
Consider 𝑙𝑜𝑔 6 𝑥. Can we find the value?
One option would be to write this as a common logarithm and use our calculator:
Let 𝑙𝑜𝑔 6 𝑥=𝑎. Then
6 𝑎 =𝑥 Take the log of both sides
𝑙𝑜𝑔 6 𝑎 = 𝑙𝑜𝑔 𝑥 What we want is a – that’s
equal to the value of 𝑙𝑜𝑔 6 𝑥
𝑎·𝑙𝑜𝑔 6 = 𝑙𝑜𝑔 𝑥 Solve for a
𝑎 = 𝑙𝑜𝑔 𝑥 𝑙𝑜𝑔 So
𝑙𝑜𝑔 6 𝑥= 𝑙𝑜𝑔 𝑥 𝑙𝑜𝑔 6

25. Graphing Calculator

26. Example
Use common logarithms to evaluate 𝑙𝑜𝑔 4 12
(Use change of base and your calculator)

27. Solution
Use common logarithms to evaluate 𝑙𝑜𝑔 4 12
(Use change of base and your calculator)
𝑙𝑜𝑔 4 12= 𝑙𝑜𝑔 12 𝑙𝑜𝑔 4
≈1.792

28. (a)
(b)
(c)
(d)

29. 𝑙𝑜𝑔 9 81 𝑥 = 𝑙𝑜𝑔 9 81 − 𝑙𝑜𝑔 9 𝑥 = 𝑙𝑜𝑔 9 9 2 − 𝑙𝑜𝑔 9 𝑥 =2− 𝑙𝑜𝑔 9 𝑥
Answer is b
(a)
(b)
(c)
(d)

30. (a)
(b)
(c)
(d)

31. (a) (b) (c) (d) 𝑙𝑜𝑔 3 27𝑦 =𝑙𝑜𝑔 3 3 𝑦 =𝑙𝑜𝑔 3+𝑙𝑜𝑔 3 𝑦 Answer is b
𝑙𝑜𝑔 3 27𝑦 =𝑙𝑜𝑔 3 3 𝑦 =𝑙𝑜𝑔 3+𝑙𝑜𝑔 3 𝑦
Answer is b
Well, actually, answer is
𝑙𝑜𝑔 𝑙𝑜𝑔 𝑦
(a)
(b)
(c)
(d)