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Properties ofLogarithms Logarithms Properties of • • How do we use properties to simplify logarithmic expressions? How do we translate between logarithms in any base? HoltMcDougal Algebra 2Algebra 2 Holt 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 Properties of Logarithms Remember that to multiply powers with the same base, you add exponents. Holt McDougal Algebra 2 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 McDougal Algebra 2 Properties of Logarithms Adding Logarithms Express as a single logarithm. Simplify, if possible. 1. log64 + log69 log6 (4 9) To add the logarithms, multiply the numbers. log6 36 = x Simplify. 2 Think: 6 x = 36. Holt McDougal Algebra 2 Properties of Logarithms Adding Logarithms Express as a single logarithm. Simplify, if possible. 2. log5625 + log525 log5 (625 • 25) To add the logarithms, multiply the numbers. log5 15,625 = x Simplify. 6 Think: 5 x = 15625. Holt McDougal Algebra 2 Properties of Logarithms Adding Logarithms Express as a single logarithm. Simplify, if possible. 1 3. log 1 27 log 1 3 3 9 1 log 1 27 9 3 log 1 3 = x To add the logarithms, multiply the numbers. Simplify. 3 –1 Holt McDougal Algebra 2 Think: 1 x 3 = 3 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 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 Properties of Logarithms Subtracting Logarithms Express as a single logarithm. Simplify, if possible. 4. log 5100 log 5 4 100 log 5 4 To subtract the logarithms, divide the numbers. log 5 25 = x Simplify. 2 Holt McDougal Algebra 2 Think: 5 x = 25. Properties of Logarithms Subtracting Logarithms Express as a single logarithm. Simplify, if possible. 5. log 7 49 log 7 7 49 log 7 7 To subtract the logarithms, divide the numbers. log 7 7 = x Simplify. 1 Holt McDougal Algebra 2 Think: 7 x = 7. Properties of Logarithms Because you can multiply logarithms, you can also take powers of logarithms. Holt McDougal Algebra 2 Properties of Logarithms Simplifying Logarithms with Exponents Express as a product. Simplify, if possible. 6. log 2 326 6 log 2 32 65 30 Holt McDougal Algebra 2 Powers expressed as multiplication. Think: 2 x = 32. Simplify. Properties of Logarithms Simplifying Logarithms with Exponents Express as a product. Simplify, if possible. 7. log 16 4 20 20 log 16 4 1 20 2 10 Holt McDougal Algebra 2 Powers expressed as multiplication. Think: 16 x = 4. Simplify. Properties of Logarithms Simplifying Logarithms with Exponents Express as a product. Simplify, if possible. 8. log10 4 4 log 10 41 4 Holt McDougal Algebra 2 Powers expressed as multiplication. Think: 10 x = 10. Simplify. Properties of Logarithms Simplifying Logarithms with Exponents Express as a product. Simplify, if possible. 9. log 5 25 2 2 log 5 25 22 4 Holt McDougal Algebra 2 Powers expressed as multiplication. Think: 5 x = 25. Simplify. Properties of Logarithms Simplifying Logarithms with Exponents Express as a product. Simplify, if possible. 1 10. log 2 2 5 1 5 log 2 2 5 1 5 Holt McDougal Algebra 2 Powers expressed as multiplication. Think: 2 x = ½ . Simplify. Properties of Logarithms Exponential and logarithmic operations undo each other since they are inverse operations. Holt McDougal Algebra 2 Properties of Logarithms Recognizing Inverses Simplify each expression. 12. log381 11 11. log33 13. 5log 10 5 log3311 log334 5log 10 11 4 10 14. log100.9 Log10100.9 0.9 Holt McDougal Algebra 2 5 15. 2log (8x) 2 2log (8x) 2 8x Properties of Logarithms Most calculators calculate logarithms only in base 10 or base e. You can change a logarithm in one base to a logarithm in another base with the following formula. Holt McDougal Algebra 2 Properties of Logarithms Changing the Base of a Logarithm 16. Evaluate log328. 17. Evaluate log927. Change to base 10 Change to base 10 log328 = log8 log32 ≈ 0.6 18. Evaluate log816. Change to base 10 log816 = log16 log8 Holt McDougal Algebra 2 ≈ 1.3 log927 = log27 log9 ≈ 1.5 Properties of Logarithms Lesson 9.1 Practice A Holt McDougal Algebra 2