4 4 Properties of Logarithms Warm Up Simplify

4 -4 Properties of Logarithms Warm Up Simplify. 1. (26)(28) 2. (3– 2)(35) 3. 4. 5. (73)5 Write in exponential form. 6. logx x = 1 Holt Mc. Dougal Algebra 2 7. 0 = logx 1

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

4 -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 Mc. Dougal Algebra 2

4 -4 Properties of Logarithms Remember that to multiply powers with the same base, you add exponents. Holt Mc. Dougal 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: log j + log a + log m = log j a m Holt Mc. Dougal Algebra 2

4 -4 Properties of Logarithms Check It Out! Example 1 a Express as a single logarithm. Simplify, if possible. log 5625 + log 525 Holt Mc. Dougal Algebra 2

4 -4 Properties of Logarithms Check It Out! Example 1 b Express as a single logarithm. Simplify, if possible. log 1 27 + log 1 3 Holt Mc. Dougal Algebra 2 3 1 9

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 Mc. Dougal Algebra 2

4 -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 Mc. Dougal Algebra 2

4 -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 Holt Mc. Dougal Algebra 2

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

4 -4 Properties of Logarithms Check It Out! Example 3 Express as a product. Simplify, if possibly. a. log 104 Holt Mc. Dougal Algebra 2 b. log 5252

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

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

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

4 -4 Properties of Logarithms Check It Out! Example 5 a Evaluate log 927. Method 1 Change to base 10. log 927 = Holt Mc. Dougal Algebra 2

4 -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. Holt Mc. Dougal Algebra 2

4 -4 Properties of Logarithms Check It Out! Example 5 b Evaluate log 816. Method 1 Change to base 10. Holt Mc. Dougal Algebra 2

4 -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. Holt Mc. Dougal Algebra 2

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 Mc. Dougal 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? Holt Mc. Dougal Algebra 2

4 -4 Properties of Logarithms Check It Out! Example 6 Continued Holt Mc. Dougal Algebra 2