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ofLogarithms Logarithms 4-4 4-4 Properties Properties of Warm Up Lesson Presentation Lesson Quiz HoltMcDougal Algebra 2Algebra 2 Holt 4-4 Properties of Logarithms Warm Up Simplify. 1. (26)(28) 214 2. (3–2)(35) 33 3. 38 44 5. (73)5 715 4. Write in exponential form. 6. logx x = 1 x1 = x Holt McDougal Algebra 2 7. 0 = logx1 x0 = 1 4-4 Properties of Logarithms Objectives Use properties to simplify logarithmic expressions. Translate between logarithms in any base. Holt McDougal Algebra 2 4-4 Properties of Logarithms The logarithmic function for pH that you saw in the previous lessons, pH =–log[H+], can also be expressed in exponential form, as 10–pH = [H+]. Because logarithms are exponents, you can derive the properties of logarithms from the properties of exponents Holt McDougal Algebra 2 4-4 Properties of Logarithms Remember that to multiply powers with the same base, you add exponents. Holt McDougal Algebra 2 4-4 Properties of Logarithms The property in the previous slide can be used in reverse to write a sum of logarithms (exponents) as a single logarithm, which can often be simplified. Helpful Hint Think: logj + loga + logm = logjam Holt McDougal Algebra 2 4-4 Properties of Logarithms Example 1: Adding Logarithms Express log64 + log69 as a single logarithm. Simplify. log64 + log69 log6 (4 9) To add the logarithms, multiply the numbers. log6 36 Simplify. 2 Think: 6? = 36. Holt McDougal Algebra 2 4-4 Properties of Logarithms Check It Out! Example 1a Express as a single logarithm. Simplify, if possible. log5625 + log525 log5 (625 • 25) To add the logarithms, multiply the numbers. log5 15,625 Simplify. 6 Think: 5? = 15625 Holt McDougal Algebra 2 4-4 Properties of Logarithms Check It Out! Example 1b Express as a single logarithm. Simplify, if possible. log 1 27 + log 1 3 3 log 1 (27 • 3 log 1 3 1 9 ) 1 9 To add the logarithms, multiply the numbers. Simplify. 3 –1 Holt McDougal Algebra 2 Think: 1 ? 3 = 3 4-4 Properties of Logarithms Remember that to divide powers with the same base, you subtract exponents Because logarithms are exponents, subtracting logarithms with the same base is the same as finding the logarithms of the quotient with that base. Holt McDougal Algebra 2 4-4 Properties of Logarithms The property above can also be used in reverse. Caution Just as a5b3 cannot be simplified, logarithms must have the same base to be simplified. Holt McDougal Algebra 2 4-4 Properties of Logarithms Example 2: Subtracting Logarithms Express log5100 – log54 as a single logarithm. Simplify, if possible. log5100 – log54 log5(100 ÷ 4) To subtract the logarithms, divide the numbers. log525 Simplify. 2 Think: 5? = 25. Holt McDougal Algebra 2 4-4 Properties of Logarithms Check It Out! Example 2 Express log749 – log77 as a single logarithm. Simplify, if possible. log749 – log77 log7(49 ÷ 7) To subtract the logarithms, divide the numbers log77 Simplify. 1 Holt McDougal Algebra 2 Think: 7? = 7. 4-4 Properties of Logarithms Because you can multiply logarithms, you can also take powers of logarithms. Holt McDougal Algebra 2 4-4 Properties of Logarithms Example 3: Simplifying Logarithms with Exponents Express as a product. Simplify, if possible. A. log2326 6log232 Because 6(5) = 30 25 = 32, log232 = 5. Holt McDougal Algebra 2 B. log8420 20log84 20( 2 3 )= 40 3 Because 2 3 8 = 4, 2 log84 = 3 . 4-4 Properties of Logarithms Check It Out! Example 3 Express as a product. Simplify, if possibly. a. log104 b. log5252 4log10 4(1) = 4 2log525 Because 101 = 10, log 10 = 1. Holt McDougal Algebra 2 2(2) = 4 Because 52 = 25, log525 = 2. 4-4 Properties of Logarithms Check It Out! Example 3 Express as a product. Simplify, if possibly. c. log2 ( 5log2 ( 1 2 1 2 )5 ) 5(–1) = –5 Holt McDougal Algebra 2 Because 1 2–1 = 2 , 1 log2 2 = –1. 4-4 Properties of Logarithms Exponential and logarithmic operations undo each other since they are inverse operations. Holt McDougal Algebra 2 4-4 Properties of Logarithms Example 4: Recognizing Inverses Simplify each expression. a. log3311 b. log381 log3311 log33 3 3 3 11 log33 4 4 Holt McDougal Algebra 2 c. 5log 10 5 5log 10 5 10 4-4 Properties of Logarithms Check It Out! Example 4 a. Simplify log100.9 b. Simplify 2log (8x) 2 log 100.9 2log (8x) 0.9 8x Holt McDougal Algebra 2 2 4-4 Properties of Logarithms 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. Holt McDougal Algebra 2 4-4 Properties of Logarithms 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. ≈ 0.6 Divide. Holt McDougal Algebra 2 4-4 Properties of Logarithms Example 5 Continued Evaluate log328. Method 2 Change to base 2, because both 32 and 8 are powers of 2. log328 = 3 = log232 5 log28 = 0.6 Holt McDougal Algebra 2 Use a calculator. 4-4 Properties of Logarithms Check It Out! Example 5a Evaluate log927. Method 1 Change to base 10. log927 = log27 log9 1.431 ≈ 0.954 Use a calculator. ≈ 1.5 Divide. Holt McDougal Algebra 2 4-4 Properties of Logarithms Check It Out! Example 5a Continued Evaluate log927. Method 2 Change to base 3, because both 27 and 9 are powers of 3. log927 = 3 = log39 2 log327 = 1.5 Holt McDougal Algebra 2 Use a calculator. 4-4 Properties of Logarithms Check It Out! Example 5b Evaluate log816. Method 1 Change to base 10. Log816 = log16 log8 1.204 ≈ 0.903 Use a calculator. ≈ 1.3 Divide. Holt McDougal Algebra 2 4-4 Properties of Logarithms 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 = 1.3 Holt McDougal Algebra 2 2 = 1.5 Use a calculator. 4-4 Properties of Logarithms 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. Helpful Hint The Richter scale is logarithmic, so an increase of 1 corresponds to a release of 10 times as much energy. Holt McDougal Algebra 2 4-4 Properties of Logarithms Example 6: Geology Application The tsunami that devastated parts of Asia in December 2004 was spawned by an earthquake with magnitude 9.3 How many times as much energy did this earthquake release compared to the 6.9-magnitude earthquake that struck San Francisco in1989? The Richter magnitude of an earthquake, M, is related to the energy released in ergs E given by the formula. Substitute 9.3 for M. Holt McDougal Algebra 2 4-4 Properties of Logarithms Example 6 Continued Multiply both sides by E 13.95 = log 11.8 10 3 2 . Simplify. Apply the Quotient Property of Logarithms. Apply the Inverse Properties of Logarithms and Exponents. Holt McDougal Algebra 2 4-4 Properties of Logarithms Example 6 Continued Given the definition of a logarithm, the logarithm is the exponent. Use a calculator to evaluate. The magnitude of the tsunami was 5.6 1025 ergs. Holt McDougal Algebra 2 4-4 Properties of Logarithms Example 6 Continued Substitute 6.9 for M. Multiply both sides by 3 2 . Simplify. Apply the Quotient Property of Logarithms. Holt McDougal Algebra 2 4-4 Properties of Logarithms Example 6 Continued Apply the Inverse Properties of Logarithms and Exponents. Given the definition of a logarithm, the logarithm is the exponent. Use a calculator to evaluate. The magnitude of the San Francisco earthquake was 1.4 1022 ergs. The tsunami released 5.6 1025 = 4000 times as 1.4 1022 much energy as the earthquake in San Francisco. Holt McDougal Algebra 2 4-4 Properties of Logarithms Check It Out! Example 6 How many times as much energy is released by an earthquake with magnitude of 9.2 by an earthquake with a magnitude of 8? Substitute 9.2 for M. Multiply both sides by Simplify. Holt McDougal Algebra 2 3 2 . 4-4 Properties of Logarithms Check It Out! Example 6 Continued Apply the Quotient Property of Logarithms. Apply the Inverse Properties of Logarithms and Exponents. Given the definition of a logarithm, the logarithm is the exponent. Use a calculator to evaluate. The magnitude of the earthquake is 4.0 1025 ergs. Holt McDougal Algebra 2 4-4 Properties of Logarithms Check It Out! Example 6 Continued Substitute 8.0 for M. Multiply both sides by Simplify. Holt McDougal Algebra 2 3 2 . 4-4 Properties of Logarithms Check It Out! Example 6 Continued Apply the Quotient Property of Logarithms. Apply the Inverse Properties of Logarithms and Exponents. Given the definition of a logarithm, the logarithm is the exponent. Use a calculator to evaluate. Holt McDougal Algebra 2 4-4 Properties of Logarithms Check It Out! Example 6 Continued The magnitude of the second earthquake was 6.3 1023 ergs. The earthquake with a magnitude 9.2 released was 4.0 1025 ≈ 63 times greater. 6.3 1023 Holt McDougal Algebra 2 4-4 Properties of Logarithms Lesson Quiz: Part I Express each as a single logarithm. 1. log69 + log624 log6216 = 3 2. log3108 – log34 log327 = 3 Simplify. 3. log2810,000 30,000 4. log44x –1 x–1 5. 10log125 125 6. log64128 Holt McDougal Algebra 2 7 6 4-4 Properties of Logarithms Lesson Quiz: Part II Use a calculator to find each logarithm to the nearest thousandth. 7. log320 2.727 8. log 1 10 –3.322 2 9. How many times as much energy is released by a magnitude-8.5 earthquake as a magntitude6.5 earthquake? 1000 Holt McDougal Algebra 2