1. 8-4 Properties of Logarithms
Hubarth
Algebra II

2. Properties of Logarithms
For any positive number 𝑀, 𝑁, and 𝑏, 𝑏≠1,
𝑙𝑜 𝑔 𝑏 𝑀𝑁=𝑙𝑜 𝑔 𝑏 𝑀+𝑙𝑜 𝑔 𝑏 𝑁 Product Property
𝑙𝑜𝑔 𝑏 𝑀 𝑁 = 𝑙𝑜𝑔 𝑏 𝑀− 𝑙𝑜𝑔 𝑏 𝑁 Quotient Property
𝑙𝑜𝑔 𝑏 𝑀 𝑥 =𝑥 𝑙𝑜𝑔 𝑏 𝑀 Power Property

3. Ex. 1 Identify the Properties of Logarithms
State the property or properties used to rewrite each expression.
𝒂. log⁡6 = log⁡2 + log⁡3
Product Property: log 6 = log (2•3) = log 2 + log 3
b. 𝑙𝑜 𝑔 𝑏 𝑥 2 𝑦 =2𝑙𝑜 𝑔 𝑏 𝑥− 𝑙𝑜𝑔 𝑏 𝑦
Quotient Property: logb = logb x2 – logb y x2 y
Power Property: 𝑙𝑜𝑔𝑏 𝑥2 – 𝑙𝑜𝑔𝑏 𝑦 = 2 𝑙𝑜𝑔𝑏 𝑥 – 𝑙𝑜𝑔𝑏 𝑦

4. Ex. 2 Simplifying Logarithms
Write each logarithmic expression as a single logarithm.
a. log4 64 – log4 16
log4 64 – log4 16 = log4 Quotient Property 64 16
= log4 4 = Simplify.
b. 6 log5 x + log5 y
6 log5 𝑥 + log5 𝑦 = log5 𝑥6 + log5 𝑦 Power Property
= log5 (𝑥6𝑦) Product Property
𝑆𝑜 log4 64 – log4 16 = log4 4, 𝑎𝑛𝑑 6 log5 𝑥 + log2 𝑦 = log5 (𝑥6𝑦).

5. Ex. 3 Expanding Logarithms
Expand each logarithm.
a. 𝑙𝑜𝑔 7 ( 𝑡 𝑢 )
= log7 t – log7 u Quotient Property
b. log⁡(4𝑝3)
= log⁡4 + log⁡𝑝3 Product Property
= log⁡4 + 3 log⁡𝑝 Power Property

6. Practice
State the property or properties used to rewrite each expression.
a. 𝑙𝑜𝑔 5 2+ 𝑙𝑜𝑔 5 6= 𝑙𝑜𝑔 b. 3 𝑙𝑜𝑔 𝑏 4−3 𝑙𝑜𝑔 𝑏 2= 𝑙𝑜𝑔 𝑏 8
Product Property
Power Property
Quotient Property
2. Write 3 log 2+𝑙𝑜𝑔4−𝑙𝑜𝑔16 as a single logarithm.
log⁡2
3. Can you write 3 𝑙𝑜𝑔 2 9− 𝑙𝑜𝑔 6 9 as a single logarithm? Explain.
No, they have different bases
4. Expand each logarithm.
a. 𝑙𝑜𝑔 2 7𝑏 b. 𝑙𝑜𝑔 ( 𝑦 3 ) 2 c. 𝑙𝑜𝑔 7 𝑎 3 𝑏 4
𝑙𝑜𝑔 2 7+ 𝑙𝑜𝑔 2 𝑏
2 log 𝑦 −2 log 3
3 𝑙𝑜𝑔 7 𝑎+4 𝑙𝑜𝑔 7 𝑏