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**Properties of Logarithms**

These properties are based on rules of exponents since logs = exponents

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**I. 𝑙𝑜𝑔 𝑏 1=0 Because in exponential form 𝑏 0 =1 𝑙𝑜𝑔 5 1= 𝑙𝑜𝑔 𝑚 1=**

(any number to the zero power = 1) 5 to what power = 1? Example: 𝑙𝑜𝑔 5 1= Example: 𝑙𝑜𝑔 𝑚 1=

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**II. 𝑙𝑜𝑔 𝑏 𝑏=1 1 1 Because in exponential form 𝑏 1 =𝑏 𝑙𝑜𝑔 5 5= 𝑙𝑜𝑔 𝑚 𝑚=**

(any number to the first power is itself) 5 to what power = 5? 1 Example: 𝑙𝑜𝑔 5 5= 1 Example: 𝑙𝑜𝑔 𝑚 𝑚=

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**III. Product Rule 𝑙𝑜𝑔 𝑏 𝑚𝑛= 𝑙𝑜𝑔 𝑏 𝑚+𝑙𝑜𝑔 𝑏 𝑛 𝑙𝑜𝑔 𝑏 𝑥𝑦 =**

𝑙𝑜𝑔 𝑏 𝑚𝑛= 𝑙𝑜𝑔 𝑏 𝑚+𝑙𝑜𝑔 𝑏 𝑛 Because in exponential form 𝑏 𝑚 ×𝑏 𝑛 = 𝑏 𝑚+𝑛 Examples: 𝑙𝑜𝑔 𝑏 𝑥𝑦 = 𝑙𝑜𝑔 𝑏 𝑥+ 𝑙𝑜𝑔 𝑏 𝑦 𝑙𝑜𝑔6 = 𝑙𝑜𝑔2+𝑙𝑜𝑔3 𝑙𝑜𝑔 3 9𝑏 = 𝑙𝑜𝑔 3 9+ 𝑙𝑜𝑔 3 𝑏

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**IV. Quotient Rule 𝑙𝑜𝑔 𝑏 𝑚 𝑛 = 𝑙𝑜𝑔 𝑏 𝑚−𝑙𝑜𝑔 𝑏 𝑛 𝑙𝑜𝑔 5 𝑥 𝑦 =**

𝑙𝑜𝑔 𝑏 𝑚 𝑛 = 𝑙𝑜𝑔 𝑏 𝑚−𝑙𝑜𝑔 𝑏 𝑛 Because in exponential form 𝑏 𝑚 𝑏 𝑛 = 𝑏 𝑚−𝑛 Examples: 𝑙𝑜𝑔 5 𝑥 𝑦 = 𝑙𝑜𝑔 5 𝑥− 𝑙𝑜𝑔 5 𝑦 𝑙𝑜𝑔 2 𝑎 3 = 𝑙𝑜𝑔 2 𝑎− 𝑙𝑜𝑔 2 3 𝑙𝑜𝑔 3 6𝑏 7 = 𝒍𝒐𝒈 𝟑 𝟔+ 𝒍𝒐𝒈 𝟑 𝒃− 𝒍𝒐𝒈 𝟑 𝟕

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**V. Power Rule 𝑙𝑜𝑔 𝑏 𝑚 𝑛 =𝑛 𝑙𝑜𝑔 𝑏 𝑚 3𝑙𝑜𝑔 2 𝑎+ 4𝑙𝑜𝑔 2 𝑏 𝑙𝑜𝑔 5 𝑥 3 =**

𝑙𝑜𝑔 𝑏 𝑚 𝑛 =𝑛 𝑙𝑜𝑔 𝑏 𝑚 Because in exponential form 𝑏 𝑚 𝑛 = 𝑏 𝑚𝑛 Examples: 𝑙𝑜𝑔 5 𝑥 3 = 3 𝑙𝑜𝑔 5 𝑥 𝑙𝑜𝑔 2 𝑎 3 𝑏 4 = 3𝑙𝑜𝑔 2 𝑎+ 4𝑙𝑜𝑔 2 𝑏

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**𝑙𝑜𝑔 𝑏 𝑚= 𝑙𝑜𝑔𝑚 𝑙𝑜𝑔𝑏 𝑙𝑜𝑔9 𝑙𝑜𝑔5 𝑙𝑜𝑔 5 9 = VI. Change of Base Formula**

Example: 𝑙𝑜𝑔 = These properties remain the same when working with the natural log.

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**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

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**Use the properties of logs to expand the following expressions:**

1. 1. Apply Product Rule: 2. Apply Power Rule:

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**Use the properties of logs to expand the following expressions:**

2. 1. Apply Product Rule: 2. Apply Power Rule:

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**Use the properties of logs to expand the following expressions:**

3. 1. Apply Quotient Rule: 2. Apply Product Rule:

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**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:

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**Use the properties of logs to expand the following expressions:**

5. 2. Apply Product Rule: 3. Apply Power Rule:

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**Write as a single logarithmic expression.**

5. 1. Apply Reverse Power Rule: 2. Apply Reverse Quotient Rule: 3. Change to radical form

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**Write as a single logarithmic expression.**

6. 1. Apply Reverse Product Rule: 2. Simplify

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**Write as a single logarithmic expression.**

1. Apply Reverse Power Rule: 6. 2. Apply Reverse Product Rule:

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Practice Time

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