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**Exponential and logarithmic operations undo each other since they are inverse operations.**

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**Logs and Exp as inverses**

WHY! Rewrite it as a logarithm

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**Example 4: Recognizing Inverses**

Simplify each expression. a. log3311 b. log381 c. 5log510 log3311 log33 3 3 3 5log510 11 log334 10 4

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Check It Out! Example 4 a. Simplify log100.9 b. Simplify 2log2(8x) log 100.9 2log2(8x) 0.9 8x

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Most calculators calculate logarithms only in base 10 or base e (see Lesson 7-6). You can change a logarithm in one base to a logarithm in another base with the following formula.

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**Example 5: Changing the Base of a Logarithm**

Evaluate log328. Method 1 Change to base 10 log328 = log8 log32 0.903 1.51 ≈ Use a calculator. Divide. ≈ 0.6

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**Method 2 Change to base 2, because both 32 and 8 are powers of 2.**

Example 5 Continued Evaluate log328. Method 2 Change to base 2, because both 32 and 8 are powers of 2. log328 = log28 log232 = 3 5 Use a calculator. = 0.6

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**Check It Out! Example 5a Evaluate log927. Method 1 Change to base 10.**

1.431 0.954 ≈ Use a calculator. ≈ 1.5 Divide.

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**Check It Out! Example 5a Continued**

Evaluate log927. Method 2 Change to base 3, because both 27 and 9 are powers of 3. log927 = log327 log39 = 3 2 Use a calculator. = 1.5

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**Check It Out! Example 5b Evaluate log816. Method 1 Change to base 10.**

1.204 0.903 ≈ Use a calculator. Divide. ≈ 1.3

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**Check It Out! Example 5b Continued**

Evaluate log816. Method 2 Change to base 4, because both 16 and 8 are powers of 2. log816 = log416 log48 = 2 1.5 Use a calculator. = 1.3

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**So…why does that formula change bases?**

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Logarithmic scales are useful for measuring quantities that have a very wide range of values, such as the intensity (loudness) of a sound or the energy released by an earthquake. The Richter scale is logarithmic, so an increase of 1 corresponds to a release of 10 times as much energy. Helpful Hint

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**Use a calculator to find each logarithm to the nearest thousandth.**

Lesson Quiz: Part II Use a calculator to find each logarithm to the nearest thousandth. 7. log320 2.727 8. log 10 1 2 –3.322

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Homework p # 32-36

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