7 4 Properties of Logarithms Objectives Use properties

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 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. 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: log j+ log a+ log m = log jam Holt Algebra 2

7 -4 Properties of 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 Express as a single logarithm. Simplify, if possible. log 5625 + log 525 Holt Algebra 2 log 1 27 + log 1 3 3 1 9

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 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 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 Express as a product. Simplify, if possible. A. log 2326 B. log 8420 6 log 232 20 log 84 6(5) = 30 20( Because 25 = 32, log 232 = 5. Holt Algebra 2 2 3 )= 40 3 Because 2 3 8 = 4, 2 log 84 = 3.

7 -4 Properties of Logarithms Express as a product. Simplify, if possibly. a. log 104 b. log 5252 c. log 2 ( 1 2 )5 4 log 10 2 log 525 5 log 2 ( 4(1) = 4 2(2) = 4 5(– 1) = – 5 Because 101 = 10, log 10 = 1. Holt Algebra 2 Because 52 = 25, log 525 = 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 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 Simplify each expression. 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 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 Changing the Base of a Logarithm Evaluate log 927. Evaluate log 816. Change to base 10. log 927 = Holt Algebra 2 log 27 log 9 Log 816 = log 16 log 8 1. 431 ≈ 0. 954 1. 204 ≈ 0. 903 ≈ 1. 5 ≈ 1. 3

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 Holt Algebra 2