1. Lesson 3.4 Properties of Logarithms
Objective: To learn and apply the properties of logarithms.

2. Think about this…
If a logarithm is the inverse of an exponential, what do you think we can surmise about the properties of logarithms?
They should be the inverse of the properties of exponents! For example, if we add exponents when we multiply in the same base, what would we do to logs when they are being multiplied?

3. PRODUCT RULE Product Property: logb(MN) = logbM + logbN
The logarithm of a product is the sum of the logarithms of the factors.
Ex) logbx3 + logby =

4. See Example 1
Express as a single logarithm:

5. Check Point 1
Use the product rule to expand each logarithmic expression:
A) log6(7 11) B) log(100x)

6. QUOTIENT RULE Quotient Property logb(M/N) = logbM – logbN
The logarithm of a quotient is the logarithm of the numerator minus the logarithm of the denominator.
Ex) log2w - log216 =

7. See Example 2
Express as a difference of logarithms.

8. Check Point 2
Use the quotient rule to expand each logarithmic expression:

9. POWER RULE Power Property: logbMp = p logbM
The logarithm of a power of M is the exponent times the logarithm of M.
Ex) log2x3 =

10. See Example 3
Express as a product.

11. Check Point 3
Use the power rule to expand each logarithmic expression:

12. Extra Practice
Express as a product.

13. Expanding Logarithmic Expressions (See blue box on page 416.)
Use properties of logarithms to change one logarithm into a sum or difference of others.
Example

14. See Example 4
Check Point 4: Use log properties to expand each expression as much as possible.

15. Expanding Logs – Express as a sum or difference.

16. More Practice Expanding
log27b
log(y/3)2
c) log7a3b4

17. Condensing Logarithmic Expressions
We can also use the properties of logarithms to condense expressions or “write as a single logarithm”.

18. Let’s reverse things.
Express as a single logarithm.

19. Pencils down. Watch and listen.
Express as a single logarithm.
Solution:

20. Check Point 5
Write as a single logarithm.

21. Check Point 6 Write as a single logarithm.

22. Check Point 6 Write as a single logarithm.

23. More Practice d) Write 3log2 + log 4 – log 16 as a single logarithm.
e) Can you write 3log29 – log69 as a single logarithm?

24. Review of Properties (from Lesson 3.2)
The Logarithm of a Base to a Power
For any base a and any real number x,
loga a x = x.
(The logarithm, base a, of a to a power is the power.)
A Base to a Logarithmic Power
For any base a and any positive real number x,
(The number a raised to the power loga x is x.)

25. Examples
Simplify.
a) loga a 6
b) ln e 8

26. Simplify. A) B)

27. Change of Base Formula
The 2 bases we are most able to calculate logarithms for are base 10 and base e. These are the only bases that our calculators have buttons for.
For ease of computing a logarithm, we may want to switch from one base to another using the formula

28. See Examples Use common logs to evaluate log7 2506.
Use natural logs to evaluate log

29. Solve:

30. Summary of Properties of Logarithms

31. Summary of Properties of Logarithms (cont.)