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

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=

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=

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

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

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

𝑙𝑜𝑔 𝑏 𝑚= 𝑙𝑜𝑔𝑚 𝑙𝑜𝑔𝑏 𝑙𝑜𝑔9 𝑙𝑜𝑔5 𝑙𝑜𝑔 5 9 = VI. Change of Base Formula Example: 𝑙𝑜𝑔 5 9 = These properties remain the same when working with the natural log.

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

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

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

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

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:

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

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

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

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

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