1. Apply Properties of logarithms Lesson 4.5
Honors Algebra 2
Apply Properties of logarithms
Lesson 4.5

2. Goals Goal Rubric Use properties of logarithms.
Expand and condense a logarithmic expression.
Use the change of base formula.
Use properties of logarithms in real life.
Level 1 – Know the goals.
Level 2 – Fully understand the goals.
Level 3 – Use the goals to solve simple problems.
Level 4 – Use the goals to solve more advanced problems.
Level 5 – Adapts and applies the goals to different and more complex problems.

3. Vocabulary
Base

4. Properties of logarithms
Because logarithms are exponents, you can derive the properties of logarithms from the properties of exponents.

5. Properties of logarithms
Remember that to multiply powers with the same base, you add exponents.
Logarithm of a product is equal to the sum of the logarithms.

6. Properties of logarithms
Using the product property as previously shown is called expanding the logarithm.
log 𝑎𝑏𝑐 = log 𝑎 + log 𝑏 + log 𝑐
The property can also be used in reverse to write a sum of logarithms as a single logarithm, called condensing the logarithm.
log 𝑎 + log 𝑏 + log 𝑐 = log 𝑎𝑏𝑐

7. Properties of logarithms
Remember that to divide powers with the same base, you subtract exponents.
Because logarithms are exponents, subtracting logarithms with the same base is the same as finding the logarithms of the quotient with that base.

8. Properties of logarithms
Logarithm of a quotient is equal to the difference of the logarithms.
The property above can also be used in reverse.

9. Properties of logarithms
Just as a5b3 cannot be simplified, logarithms must have the same base to be simplified.
Caution

10. Properties of logarithms
Because you can multiply logarithms, you can also take powers of logarithms.
Logarithm of a power is the product of the power and the logarithm.

11. Summary Properties of Logarithms
Product Property: Logarithm of a product is equal to the sum of the logarithms.
Quotient Property: Logarithm of a quotient is equal to the difference of the logarithms.
Power Property: Logarithm of a power is the product of the power and the logarithm.

12. The properties of logarithms are useful for rewriting logarithmic expressions in forms that simplify the operations of algebra.
This is because the properties convert more complicated products, quotients, and exponential forms into simpler sums, differences, and products.
This is called expanding a logarithmic expression.
The procedure above can be reversed to produce a single logarithmic expression.
This is called condensing a logarithmic expression.

13. Use properties of logarithms
EXAMPLE 1
Use properties of logarithms 3 4
log
Use
0.792 and 7
1.404 to evaluate the
logarithm. a. 4
log 3 7 = 3 – 4
log 7
Quotient property
0.792
1.404 –
Use the given values of 3 4
log 7.
and =
–0.612
Simplify. b. 4
log 21 = 4
log
(3 7)
Write 21 as = 3 4
log + 7
Product property
0.792
1.404 +
Use the given values of 3 4
log 7.
and =
2.196
Simplify.

14. Use properties of logarithms
EXAMPLE 1
Use properties of logarithms 3 4
log
Use
0.792 and 7
1.404 to evaluate the
logarithm. c. 4
log 49 72 = 4
log
Write 49 as 72 4
log = 2 7
Power property
2(1.404)
Use the given value of 7. 4
log =
2.808
Simplify.

15. Your Turn:
for Example 1 5 6
log
Use
0.898 and 8
1.161 to evaluate the
logarithm. 1. 5 8 6
log
–0.263
ANSWER 2. 6
log 40
2.059
ANSWER

16. Your Turn:
for Example 1 5 6
log
Use
0.898 and 8
1.161 to evaluate the
logarithm. 6
log 3. 64
2.322
ANSWER 4. 6
log
125
2.694
ANSWER

17. EXAMPLE 2 Expand a logarithmic expression Expand 6 log 5x3 y SOLUTION=
5x3 y 6
log –
Quotient property = 5 6
log x3 y – +
Product property = 5 6
log x y – + 3
Power property

18. Condense a logarithmic expression
EXAMPLE 3
Condense a logarithmic
expression
SOLUTION –
log 9 + 3log2
log 3 = –
log 9 + log 23
log 3
Power property =
log
( ) 23 –
log 3
Product property =
log 9 23 3
Quotient property = 24
log
Simplify.
The correct answer is D.
ANSWER

19. Your Turn:
for Examples 2 and 3
Expand 5.
log 3 x4 .
log log x
ANSWER

20. Your Turn:
for Examples 2 and 3
Condense ln ln 3 – ln 12. 6.
ln 9
ANSWER

21. Change of Base Formula
Most calculators calculate logarithms only in base 10 or base e. You can change a logarithm in one base to a logarithm in another base with the following formula.

22. Change of Base Formula Change to base 10: log 𝑏 𝑥 = log 𝑥 log 𝑏
Change to base e : log 𝑏 𝑥 = ln 𝑥 ln 𝑏

23. EXAMPLE 4
Use the change-of-base formula 3
log 8
Evaluate
using common logarithms and natural
logarithms.
SOLUTION
Using common logarithms: =
log 8
log 3
0.9031
0.4771 3
log 8
1.893
Using natural logarithms: =
ln 8
ln 3
2.0794
1.0986 3
log 8
1.893

24. Your Turn:
for Example 4
Use the change-of-base formula to evaluate the logarithm. 5
log 8 7.
about 1.292
ANSWER 8
log 14 8.
about 1.269
ANSWER

25. Your Turn:
for Example 4
Use the change-of-base formula to evaluate the logarithm. 26
log 9 9.
about 0.674
ANSWER
10. 12
log 30
about 1.369
ANSWER

26. EXAMPLE 5
Use properties of logarithms
in real life
Sound Intensity
For a sound with intensity I (in watts per square meter), the loudness L(I) of the sound (in decibels) is given by the function =
log
L(I) 10 I I
where is the intensity of a barely audible sound (about watts per square meter). An artist in a recording studio turns up the volume of a track so that the sound’s intensity doubles. By how many decibels does the loudness increase?
10–12

27. Use properties of logarithms in real life
EXAMPLE 5
Use properties of logarithms
in real life
SOLUTION
Let I be the original intensity, so that 2I is the doubled intensity.
Increase in loudness =
L(2I) – L(I)
Write an expression. =
log 10 I 2I –
Substitute. = 10
log – I 2I
Distributive property = 2 10
log I – +
Product property 10
log 2 =
Simplify.
3.01
Use a calculator.
ANSWER
The loudness increases by about 3 decibels.

28. Your Turn:
for Example 5
WHAT IF? In Example 5, suppose the artist turns up the volume so that the sound’s intensity triples. By how many decibels does the loudness increase?
11.
ANSWER
The loudness increases by about decibels.