Using Properties of Logarithms

Example Use the product rule to expand each logarithmic expression. 𝑙𝑜𝑔 3 9·5 𝑙𝑜𝑔 1000𝑥

Solution Use the product rule to expand each logarithmic expression. 𝑙𝑜𝑔 3 9·5 = 𝑙𝑜𝑔 3 9 · 𝑙𝑜𝑔 3 5 𝑙𝑜𝑔 1000𝑥 =𝑙𝑜𝑔 1000 +𝑙𝑜𝑔 𝑥

Example Use the quotient rule to expand each logarithmic expression. 𝑙𝑜𝑔 5 25 𝑥 𝑙𝑜𝑔 𝑥 8 𝑙𝑛 𝑒 3 7

Solution Use the quotient rule to expand each logarithmic expression. 𝑙𝑜𝑔 5 25 𝑥 = 𝑙𝑜𝑔 5 25 − 𝑙𝑜𝑔 5 𝑥 𝑙𝑜𝑔 𝑥 8 =𝑙𝑜𝑔 𝑥 −𝑙𝑜𝑔 8 𝑙𝑛 𝑒 3 7 =𝑙𝑛 𝑒 3 −𝑙𝑛 7

Example Use the power rule to expand each logarithmic expression. 𝑙𝑜𝑔 5 7 2 𝑙𝑜𝑔 2 8𝑥 4 𝑙𝑜𝑔 𝑥 𝑙𝑛 6𝑒 5

Solution Use the power rule to expand each logarithmic expression. 𝑙𝑜𝑔 5 7 2 =2 𝑙𝑜𝑔 5 7 𝑙𝑜𝑔 2 8𝑥 4 =4 𝑙𝑜𝑔 2 8𝑥 𝑙𝑜𝑔 𝑥 = 1 2 𝑙𝑜𝑔𝑥 𝑙𝑛 6𝑒 5 =5𝑙𝑛 6𝑒

Expanding Logarithmic Expressions

Study Tip

Example Use logarithmic properties to expand each expression as much as possible. 𝑙𝑜𝑔 𝑏 𝑥 𝑧 2 𝑙𝑜𝑔 5 25 𝑦 3 𝑙𝑛 10 𝑒 𝑙𝑜𝑔 9 10

Solutions Use logarithmic properties to expand each expression as much as possible. 𝑙𝑜𝑔 𝑏 𝑥 𝑧 2 =2 𝑙𝑜𝑔 𝑏 𝑥+ 1 2 𝑙𝑜𝑔 𝑏 𝑧 =2 𝑙𝑜𝑔 𝑏 𝑥+ 𝑙𝑜𝑔 𝑏 𝑧 𝑙𝑜𝑔 5 25 𝑦 3 =3 𝑙𝑜𝑔 5 25− 𝑙𝑜𝑔 5 𝑦 =3 2− 𝑙𝑜𝑔 5 𝑦 =6−3 𝑙𝑜𝑔 5 𝑦 𝑙𝑛 10 𝑒 =𝑙𝑛 10 −𝑙𝑛 𝑒 =𝑙𝑛 10 −1 𝑙𝑜𝑔 9 10 =𝑙𝑜𝑔 9 + 1 2 𝑙𝑜𝑔 10 =𝑙𝑜𝑔 9 + 1 2

Condensing Logarithmic Expressions

Example Write as a single logarithm (condense) 𝑙𝑜𝑔 2 4+ 𝑙𝑜𝑔 2 8 𝑙𝑜𝑔 6𝑥 −𝑙𝑜𝑔 6 2 𝑙𝑜𝑔 3 9− 𝑙𝑜𝑔 3 27 𝑙𝑛 𝑥−2 +5𝑙𝑛 𝑥

Solution Write as a single logarithm (condense) 𝑙𝑜𝑔 2 4+ 𝑙𝑜𝑔 2 8 = 𝑙𝑜𝑔 2 4 8 𝑙𝑜𝑔 6𝑥 −𝑙𝑜𝑔 6=𝑙𝑜𝑔 6𝑥 6 =𝑙𝑜𝑔 𝑥 2 𝑙𝑜𝑔 3 9− 𝑙𝑜𝑔 3 27= 𝑙𝑜𝑔 3 9 2 27 = 𝑙𝑜𝑔 3 3 =1 𝑙𝑛 𝑥−2 +5𝑙𝑛 𝑥=𝑙𝑛 𝑥−2 · 𝑥 5 =𝑙𝑛 𝑥 5 𝑥−2

Question Can you simplify this any further? 𝑙𝑜𝑔 2 4+ 𝑙𝑜𝑔 2 8 = 𝑙𝑜𝑔 2 4 8

Question Can you simplify this any further? 𝑙𝑜𝑔 2 4+ 𝑙𝑜𝑔 2 8 = 𝑙𝑜𝑔 2 4 8 How about this: 𝑙𝑜𝑔 2 4 + 𝑙𝑜𝑔 2 8 = 𝑙𝑜𝑔 2 2 2 + 𝑙𝑜𝑔 2 8 1 2 = 𝑙𝑜𝑔 2 2 2 + 𝑙𝑜𝑔 2 2 3 2 =2+ 3 2 = 7 2

The Change-of-Base Property

The Change-of-Base Property Consider 𝑙𝑜𝑔 6 𝑥. Can we find the value? One option would be to write this as a common logarithm and use our calculator: Let 𝑙𝑜𝑔 6 𝑥=𝑎. Then 6 𝑎 =𝑥 Take the log of both sides 𝑙𝑜𝑔 6 𝑎 = 𝑙𝑜𝑔 𝑥 What we want is a – that’s equal to the value of 𝑙𝑜𝑔 6 𝑥 𝑎·𝑙𝑜𝑔 6 = 𝑙𝑜𝑔 𝑥 Solve for a 𝑎 = 𝑙𝑜𝑔 𝑥 𝑙𝑜𝑔 6 So 𝑙𝑜𝑔 6 𝑥= 𝑙𝑜𝑔 𝑥 𝑙𝑜𝑔 6

Graphing Calculator

Example Use common logarithms to evaluate 𝑙𝑜𝑔 4 12 (Use change of base and your calculator)

Solution Use common logarithms to evaluate 𝑙𝑜𝑔 4 12 (Use change of base and your calculator) 𝑙𝑜𝑔 4 12= 𝑙𝑜𝑔 12 𝑙𝑜𝑔 4 ≈1.792

(a) (b) (c) (d)

𝑙𝑜𝑔 9 81 𝑥 = 𝑙𝑜𝑔 9 81 − 𝑙𝑜𝑔 9 𝑥 = 𝑙𝑜𝑔 9 9 2 − 𝑙𝑜𝑔 9 𝑥 =2− 𝑙𝑜𝑔 9 𝑥 Answer is b (a) (b) (c) (d)

(a) (b) (c) (d)

(a) (b) (c) (d) 𝑙𝑜𝑔 3 27𝑦 =𝑙𝑜𝑔 3 3 𝑦 =𝑙𝑜𝑔 3+𝑙𝑜𝑔 3 𝑦 Answer is b 𝑙𝑜𝑔 3 27𝑦 =𝑙𝑜𝑔 3 3 𝑦 =𝑙𝑜𝑔 3+𝑙𝑜𝑔 3 𝑦 Answer is b Well, actually, answer is 𝑙𝑜𝑔 3+ 1 3 𝑙𝑜𝑔 𝑦 (a) (b) (c) (d)