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Properties of Logarithms

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Presentation on theme: "Properties of Logarithms"β€” Presentation transcript:

1 Properties of Logarithms
These properties are based on rules of exponents since logs = exponents

2 I. π‘™π‘œπ‘” 𝑏 1=0 Because in exponential form 𝑏 0 =1 π‘™π‘œπ‘” 5 1= π‘™π‘œπ‘” π‘š 1=
(any number to the zero power = 1) 5 to what power = 1? Example: π‘™π‘œπ‘” 5 1= Example: π‘™π‘œπ‘” π‘š 1=

3 II. π‘™π‘œπ‘” 𝑏 𝑏=1 1 1 Because in exponential form 𝑏 1 =𝑏 π‘™π‘œπ‘” 5 5= π‘™π‘œπ‘” π‘š π‘š=
(any number to the first power is itself) 5 to what power = 5? 1 Example: π‘™π‘œπ‘” 5 5= 1 Example: π‘™π‘œπ‘” π‘š π‘š=

4 III. Product Rule π‘™π‘œπ‘” 𝑏 π‘šπ‘›= π‘™π‘œπ‘” 𝑏 π‘š+π‘™π‘œπ‘” 𝑏 𝑛 π‘™π‘œπ‘” 𝑏 π‘₯𝑦 =
π‘™π‘œπ‘” 𝑏 π‘šπ‘›= π‘™π‘œπ‘” 𝑏 π‘š+π‘™π‘œπ‘” 𝑏 𝑛 Because in exponential form 𝑏 π‘š ×𝑏 𝑛 = 𝑏 π‘š+𝑛 Examples: π‘™π‘œπ‘” 𝑏 π‘₯𝑦 = π‘™π‘œπ‘” 𝑏 π‘₯+ π‘™π‘œπ‘” 𝑏 𝑦 π‘™π‘œπ‘”6 = π‘™π‘œπ‘”2+π‘™π‘œπ‘”3 π‘™π‘œπ‘” 3 9𝑏 = π‘™π‘œπ‘” 3 9+ π‘™π‘œπ‘” 3 𝑏

5 IV. Quotient Rule π‘™π‘œπ‘” 𝑏 π‘š 𝑛 = π‘™π‘œπ‘” 𝑏 π‘šβˆ’π‘™π‘œπ‘” 𝑏 𝑛 π‘™π‘œπ‘” 5 π‘₯ 𝑦 =
π‘™π‘œπ‘” 𝑏 π‘š 𝑛 = π‘™π‘œπ‘” 𝑏 π‘šβˆ’π‘™π‘œπ‘” 𝑏 𝑛 Because in exponential form 𝑏 π‘š 𝑏 𝑛 = 𝑏 π‘šβˆ’π‘› Examples: π‘™π‘œπ‘” 5 π‘₯ 𝑦 = π‘™π‘œπ‘” 5 π‘₯βˆ’ π‘™π‘œπ‘” 5 𝑦 π‘™π‘œπ‘” 2 π‘Ž 3 = π‘™π‘œπ‘” 2 π‘Žβˆ’ π‘™π‘œπ‘” 2 3 π‘™π‘œπ‘” 3 6𝑏 7 = π’π’π’ˆ πŸ‘ πŸ”+ π’π’π’ˆ πŸ‘ π’ƒβˆ’ π’π’π’ˆ πŸ‘ πŸ•

6 V. Power Rule π‘™π‘œπ‘” 𝑏 π‘š 𝑛 =𝑛 π‘™π‘œπ‘” 𝑏 π‘š 3π‘™π‘œπ‘” 2 π‘Ž+ 4π‘™π‘œπ‘” 2 𝑏 π‘™π‘œπ‘” 5 π‘₯ 3 =
π‘™π‘œπ‘” 𝑏 π‘š 𝑛 =𝑛 π‘™π‘œπ‘” 𝑏 π‘š Because in exponential form 𝑏 π‘š 𝑛 = 𝑏 π‘šπ‘› Examples: π‘™π‘œπ‘” 5 π‘₯ 3 = 3 π‘™π‘œπ‘” 5 π‘₯ π‘™π‘œπ‘” 2 π‘Ž 3 𝑏 4 = 3π‘™π‘œπ‘” 2 π‘Ž+ 4π‘™π‘œπ‘” 2 𝑏

7 π‘™π‘œπ‘” 𝑏 π‘š= π‘™π‘œπ‘”π‘š π‘™π‘œπ‘”π‘ π‘™π‘œπ‘”9 π‘™π‘œπ‘”5 π‘™π‘œπ‘” 5 9 = VI. Change of Base Formula
Example: π‘™π‘œπ‘” = These properties remain the same when working with the natural log.

8 True or False: True False True True False False False False True True
Use properties of logarithms to determine if each of the following is true or false. Check your answers using your calculator True or False: True False True True False False False False True True True True

9 Use the properties of logs to expand the following expressions:
1. 1. Apply Product Rule: 2. Apply Power Rule:

10 Use the properties of logs to expand the following expressions:
2. 1. Apply Product Rule: 2. Apply Power Rule:

11 Use the properties of logs to expand the following expressions:
3. 1. Apply Quotient Rule: 2. Apply Product Rule:

12 Use the properties of logs to expand the following expressions:
4. 1. Change radical to exponential form: 2. Apply Product Rule: 3. Apply Power Rule:

13 Use the properties of logs to expand the following expressions:
5. 2. Apply Product Rule: 3. Apply Power Rule:

14 Write as a single logarithmic expression.
5. 1. Apply Reverse Power Rule: 2. Apply Reverse Quotient Rule: 3. Change to radical form

15 Write as a single logarithmic expression.
6. 1. Apply Reverse Product Rule: 2. Simplify

16 Write as a single logarithmic expression.
1. Apply Reverse Power Rule: 6. 2. Apply Reverse Product Rule:

17 Practice Time


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