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**Properties of Logarithms**

Check for Understanding – – Prove basic properties of logarithms using properties of exponents and apply those properties to solve problems. Check for Understanding – – Know that the logarithm and exponential functions are inverses and use this information to solve real-world problems.

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Since logarithms are exponents, the properties of logarithms are similar to the properties of exponents.

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**n Product Property logb mn = logb m + logb n Quotient Property**

logb m = logb m – logb n n Power Property logb mp = p logb m m > 0, n > 0, b > 0, b ≠ 1

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Use log2 3 ≈ , log2 5 ≈ , and log2 7 ≈ to approximate the value of each expression. log2 35 log2 7 ∙ 5 log2 7 + log2 5 5.1293

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**2. log2 45 log2 32 ∙ 5 log2 32 + log2 5 2log2 3 + log2 5**

Use log2 3 ≈ , log2 5 ≈ , and log2 7 ≈ to approximate the value of each expression. 2. log2 45 log2 32 ∙ 5 log log2 5 2log2 3 + log2 5 2(1.5850) 5.4919

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**3. log2 4.2 log2 (3 ∙ 7) ÷ 5 log2 3 + log2 7 – log2 5**

Use log2 3 ≈ , log2 5 ≈ , and log2 7 ≈ to approximate the value of each expression. 3. log2 4.2 log2 (3 ∙ 7) ÷ 5 log2 3 + log2 7 – log2 5 – 2.0705

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**4. log5 2x – log5 3 = log5 8 log5 = log5 8 = 8 2x = 24 x = 12**

Solve each equation. Check your solutions. 4. log5 2x – log5 3 = log5 8 log = log5 8 = 8 2x = 24 x = 12

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**5. log2 (x + 1) + log2 5 = log2 80 – log2 4 log2 5(x + 1)= log2 20**

Solve each equation. Check your solutions. 5. log2 (x + 1) + log2 5 = log2 80 – log2 4 log2 5(x + 1)= log2 20 5x + 5 = 20 5x = 15 x = 3

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**3log2 x – 2log2 5x = 2 100x2 = x3 log2 x3 – log2 (5x)2 = 2**

Solve each equation. Check your solutions. 3log2 x – 2log2 5x = 2 log2 x3 – log2 (5x)2 = 2 100x2 = x3 0 = x3 – 100x2 0 = x2(x – 100) 0 = x = x – 100 log = 2 22 = x = x = 100 4 =

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**½ log6 25 + log6 x = log6 20 8. log7 x + 2log7 x – log7 3 = log7 72**

Solve each equation. Check your solutions. ½ log log6 x = log6 20 8. log7 x + 2log7 x – log7 3 = log7 72

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**½ log6 25 + log6 x = log6 20 4 log7 x + 2log7 x – log7 3 = log7 72 6**

Solve each equation. Check your solutions. ½ log log6 x = log6 20 4 log7 x + 2log7 x – log7 3 = log7 72 6

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