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EXAMPLE 4

Evaluate. 8. 8. 8. using common logarithms and natural. logarithms. log. log. log. 3. 3. 3. log 8. 0.9031. 1.893. =. 0.4771. log 3. ln 8. 2.0794. 1.893. =. 1.0986. ln 3. EXAMPLE 4. Use the change-of-base formula. SOLUTION. Using common logarithms:.

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EXAMPLE 4

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  1. Evaluate 8 8 8 using common logarithms and natural logarithms. log log log 3 3 3 log 8 0.9031 1.893 = 0.4771 log 3 ln 8 2.0794 1.893 = 1.0986 ln 3 EXAMPLE 4 Use the change-of-base formula SOLUTION Using common logarithms: Using natural logarithms:

  2. Sound Intensity I I I L(I) 10 log = 0 0 where is the intensity of a barely audible sound (about watts per square meter). An artist in a recording studio turns up the volume of a track so that the sound’s intensity doubles. By how many decibels does the loudness increase? 10–12 EXAMPLE 5 Use properties of logarithms in real life For a sound with intensity I (in watts per square meter), the loudness L(I) of the sound (in decibels) is given by the function

  3. L(2I) – L(I) = – 10 log 10 log = 2I 2I I I I I I I I I I I 0 0 0 0 0 0 10 – log log = – log 10 log 2 log = + 10 log 2 = 3.01 ANSWER The loudness increases by about 3decibels. EXAMPLE 5 Use properties of logarithms in real life SOLUTION Let Ibe the original intensity, so that 2Iis the doubled intensity. Increase in loudness Write an expression. Substitute. Distributive property Product property Simplify. Use a calculator.

  4. 8 14 8 14 log log log log 8 8 5 5 7. log 8 0.9031 1.292 = 0.6989 log 5 8. log 14 1.146 1.269 = 0.9031 log 8 for Examples 4 and 5 GUIDED PRACTICE Use the change-of-base formula to evaluate the logarithm. SOLUTION SOLUTION

  5. log log log log 12 26 26 12 9. 30 9 30 9 log 9 0.9542 0.674 = 1.4149 log 26 10. log 30 1.4777 1.369 = 1.076 log 12 for Examples 4 and 5 GUIDED PRACTICE Use the change-of-base formula to evaluate the logarithm. SOLUTION SOLUTION

  6. WHAT IF?In Example 5, suppose the artist turns up the volume so that the sound’s intensity triples. By how many decibels does the loudness increase? 11. I I L(I) 10 log = 0 for Examples 4 and 5 GUIDED PRACTICE SOLUTION Let Ibe the original intensity, so that 3Iis the tripled intensity.

  7. Increase in loudness L(3I) – L(I) = – 10 log 10 log = – 10 log log = 3I 3I I I I I – log 10 log 3 log = + I I I I I I 10 0 0 0 0 0 0 log 3 = 4.771 ANSWER The loudness increases by about 4.771decibels. for Examples 4 and 5 GUIDED PRACTICE Write an expression. Substitute. Distributive property Product property Simplify. Use a calculator.

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