of Logarithms 7 4 Properties of Warm Up

of. Logarithms 7 -4 Properties of Warm Up Lesson Presentation Lesson Quiz Holt Algebra 22

7 -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 x 1 = x Holt Algebra 2 7. 0 = logx 1 x 0 = 1

7 -4 Properties of Logarithms Objectives Use properties to simplify logarithmic expressions. Translate between logarithms in any base. Holt Algebra 2

7 -4 Properties of Logarithms The logarithmic function for p. H that you saw in the previous lessons, p. H =–log[H+], can also be expressed in exponential form, as 10–p. H = [H+]. Because logarithms are exponents, you can derive the properties of logarithms from the properties of exponents Holt Algebra 2

7 -4 Properties of Logarithms Remember that to multiply powers with the same base, you add exponents. Holt Algebra 2

7 -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 Algebra 2

7 -4 Properties of Logarithms Example 1: Adding Logarithms Express log 64 + log 69 as a single logarithm. Simplify. log 64 + log 69 log 6 (4 9) To add the logarithms, multiply the numbers. log 6 36 Simplify. 2 Think: 6? = 36. Holt Algebra 2

7 -4 Properties of Logarithms Check It Out! Example 1 a Express as a single logarithm. Simplify, if possible. log 5625 + log 525 log 5 (625 • 25) To add the logarithms, multiply the numbers. log 5 15, 625 Simplify. 6 Think: 5? = 15625 Holt Algebra 2

7 -4 Properties of Logarithms Check It Out! Example 1 b Express as a single logarithm. Simplify, if possible. log 1 27 + log 1 3 3 log 1 (27 • 3 1 9 ) 1 9 To add the logarithms, multiply the numbers. log 1 3 Simplify. – 1 Think: 3 Holt Algebra 2 1 ? 3 = 3

7 -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 Algebra 2

7 -4 Properties of Logarithms The property above can also be used in reverse. Caution Just as a 5 b 3 cannot be simplified, logarithms must have the same base to be simplified. Holt Algebra 2

7 -4 Properties of Logarithms Example 2: Subtracting Logarithms Express log 5100 – log 54 as a single logarithm. Simplify, if possible. log 5100 – log 54 log 5(100 ÷ 4) To subtract the logarithms, divide the numbers. log 525 Simplify. 2 Think: 5? = 25. Holt Algebra 2

7 -4 Properties of Logarithms Check It Out! Example 2 Express log 749 – log 77 as a single logarithm. Simplify, if possible. log 749 – log 77 log 7(49 ÷ 7) To subtract the logarithms, divide the numbers log 77 Simplify. 1 Holt Algebra 2 Think: 7? = 7.

7 -4 Properties of Logarithms Because you can multiply logarithms, you can also take powers of logarithms. Holt Algebra 2

7 -4 Properties of Logarithms Example 3: Simplifying Logarithms with Exponents Express as a product. Simplify, if possible. A. log 2326 6 log 232 Because 6(5) = 30 25 = 32, log 232 = 5. Holt Algebra 2 B. log 8420 20 log 84 20( 2 3 )= 40 3 Because 2 3 8 = 4, 2 log 84 = 3.

7 -4 Properties of Logarithms Check It Out! Example 3 Express as a product. Simplify, if possibly. a. log 104 b. log 5252 4 log 10 4(1) = 4 Holt Algebra 2 2 log 525 Because 101 = 10, log 10 = 1. 2(2) = 4 Because 52 = 25, log 525 = 2.

7 -4 Properties of Logarithms Check It Out! Example 3 Express as a product. Simplify, if possibly. c. log 2 ( 5 log 2 ( 1 2 )5 ) 5(– 1) = – 5 Holt Algebra 2 Because 1 2– 1 = 2 , 1 log 2 2 = – 1.

7 -4 Properties of Logarithms Exponential and logarithmic operations undo each other since they are inverse operations. Holt Algebra 2

7 -4 Properties of Logarithms Example 4: Recognizing Inverses Simplify each expression. a. log 3311 b. log 381 c. 5 log 10 5 log 3311 log 33 3 5 log 10 11 log 334 10 4 Holt Algebra 2 5

7 -4 Properties of Logarithms Check It Out! Example 4 a. Simplify log 100. 9 b. Simplify 2 log (8 x) 2 log 100. 9 2 log (8 x) 0. 9 8 x Holt Algebra 2 2

7 -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 Algebra 2

7 -4 Properties of Logarithms Example 5: Changing the Base of a Logarithm Evaluate log 328. Method 1 Change to base 10 log 328 = Holt Algebra 2 log 8 log 32 0. 903 ≈ 1. 51 Use a calculator. ≈ 0. 6 Divide.

7 -4 Properties of Logarithms Example 5 Continued Evaluate log 328. Method 2 Change to base 2, because both 32 and 8 are powers of 2. log 328 = log 28 log 232 = 0. 6 Holt Algebra 2 = 3 5 Use a calculator.

7 -4 Properties of Logarithms Check It Out! Example 5 a Evaluate log 927. Method 1 Change to base 10. log 927 = Holt Algebra 2 log 27 log 9 1. 431 ≈ 0. 954 Use a calculator. ≈ 1. 5 Divide.

7 -4 Properties of Logarithms Check It Out! Example 5 a Continued Evaluate log 927. Method 2 Change to base 3, because both 27 and 9 are powers of 3. log 927 = log 327 log 39 = 1. 5 Holt Algebra 2 = 3 2 Use a calculator.

7 -4 Properties of Logarithms Check It Out! Example 5 b Evaluate log 816. Method 1 Change to base 10. Log 816 = Holt Algebra 2 log 16 log 8 1. 204 ≈ 0. 903 Use a calculator. ≈ 1. 3 Divide.

7 -4 Properties of Logarithms Check It Out! Example 5 b Continued Evaluate log 816. Method 2 Change to base 4, because both 16 and 8 are powers of 2. log 816 = log 416 log 48 = 1. 3 Holt Algebra 2 2 = 1. 5 Use a calculator.

7 -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 Algebra 2

7 -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 in 1989? 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 Algebra 2

7 -4 Properties of Logarithms Example 6 Continued Multiply both sides by æ E ö 13. 95 = log ç 11. 8 ÷ è 10 ø 3 2 . Simplify. Apply the Quotient Property of Logarithms. Apply the Inverse Properties of Logarithms and Exponents. Holt Algebra 2

7 -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 Algebra 2

7 -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 Algebra 2

7 -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 1. 4 1022 = 4000 times as much energy as the earthquake in San Francisco. Holt Algebra 2

7 -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 Algebra 2 3 2 .

7 -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 Algebra 2

7 -4 Properties of Logarithms Check It Out! Example 6 Continued Substitute 8. 0 for M. Multiply both sides by Simplify. Holt Algebra 2 3 2 .

7 -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 Algebra 2

7 -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 Holt Algebra 2 4. 0 1025 6. 3 1023 ≈ 63 times greater.

7 -4 Properties of Logarithms Lesson Quiz: Part I Express each as a single logarithm. 1. log 69 + log 624 log 6216 = 3 2. log 3108 – log 34 log 327 = 3 Simplify. 3. log 2810, 000 30, 000 4. log 44 x – 1 x– 1 5. 10 log 125 6. log 64128 Holt Algebra 2 7 6

7 -4 Properties of Logarithms Lesson Quiz: Part II Use a calculator to find each logarithm to the nearest thousandth. 7. log 320 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 magntitude 6. 5 earthquake? 1000 Holt Algebra 2