7-4 Properties of Logarithms Warm Up Lesson Presentation Lesson Quiz Holt Algebra 2

Warm Up Simplify. 1. (26)(28) 214 2. (3–2)(35) 33 3. 38 4. 44 5. (73)5 715 Write in exponential form. 6. logx x = 1 7. 0 = logx1 x1 = x x0 = 1

Objectives “I can…” Use properties to simplify logarithmic expressions. Translate between logarithms in any base.

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

Remember that to multiply powers with the same base, you add exponents.

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

Example 1: Adding Logarithms Express log64 + log69 as a single logarithm. Simplify. log64 + log69 To add the logarithms, multiply the numbers. log6 (4  9) log6 36 Simplify. 2 Think: 6? = 36.

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

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

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.

The property above can also be used in reverse. Just as a5b3 cannot be simplified, logarithms must have the same base to be simplified. Caution

Example 2: Subtracting Logarithms Express log5100 – log54 as a single logarithm. Simplify, if possible. log5100 – log54 To subtract the logarithms, divide the numbers. log5(100 ÷ 4) log525 Simplify. 2 Think: 5? = 25.

Check It Out! Example 2 Express log749 – log77 as a single logarithm. Simplify, if possible. log749 – log77 To subtract the logarithms, divide the numbers log7(49 ÷ 7) log77 Simplify. 1 Think: 7? = 7.

Because you can multiply logarithms, you can also take powers of logarithms.

Example 3: Simplifying Logarithms with Exponents Express as a product. Simplify, if possible. A. log2326 B. log8420 6log232 20log84 Because 25 = 32, log232 = 5. Because 8 = 4, log84 = . 2 3 6(5) = 30 20( ) = 40 3 2

Check It Out! Example 3 Express as a product. Simplify, if possibly. a. log104 b. log5252 4log10 2log525 Because 101 = 10, log 10 = 1. Because 52 = 25, log525 = 2. 4(1) = 4 2(2) = 4

Express as a product. Simplify, if possibly. Check It Out! Example 3 Express as a product. Simplify, if possibly. c. log2 ( )5 1 2 5log2 ( ) 1 2 Because 2–1 = , log2 = –1. 1 2 5(–1) = –5

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

Example 4: Recognizing Inverses Simplify each expression. a. log3311 b. log381 c. 5log510 log3311 log33  3  3  3 5log510 11 log334 10 4

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

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.

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

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

Check It Out! Example 5a Evaluate log927. Method 1 Change to base 10. 1.431 0.954 ≈ Use a calculator. ≈ 1.5 Divide.

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

Check It Out! Example 5b Evaluate log816. Method 1 Change to base 10. 1.204 0.903 ≈ Use a calculator. Divide. ≈ 1.3

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

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

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.

Apply the Quotient Property of Logarithms. Example 6 Continued Multiply both sides by . 3 2 11.8 13.95 = log 10 E æ ç è ö ÷ ø Simplify. Apply the Quotient Property of Logarithms. Apply the Inverse Properties of Logarithms and Exponents.

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.

Apply the Quotient Property of Logarithms. Example 6 Continued Substitute 6.9 for M. Multiply both sides by . 3 2 Simplify. Apply the Quotient Property of Logarithms.

Apply the Inverse Properties of Logarithms and Exponents. 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 = 4000 times as much energy as the earthquake in San Francisco. 1.4  1022 5.6  1025

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 . 3 2 Simplify.

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.

Check It Out! Example 6 Continued Substitute 8.0 for M. Multiply both sides by . 3 2 Simplify.

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.

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 ≈ 63 times greater. 6.3  1023 4.0  1025

Express each as a single logarithm. 1. log69 + log624 log6216 = 3 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 7 6

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 9. How many times as much energy is released by a magnitude-8.5 earthquake as a magntitude-6.5 earthquake? 1000