WarmUp Exercises Evaluate the logarithm 1 log 5

Warm-Up Exercises Evaluate the logarithm. 1. log 5 625 ANSWER 4 2. log 0. 00001 ANSWER – 5

Warm-Up Exercises Evaluate the logarithm. 3. log 32 2 ANSWER 1 5 1 4. log 36 6 ANSWER – 1 2

Warm-Up Exercises Evaluate the logarithm. 5. log 8 4 ANSWER 2 3

Warm-Up Exercises

Warm-Up 1 Exercises EXAMPLE Use properties of logarithms Use log 4 3 0. 792 and log 47 1. 404 to evaluate the logarithm. a. log 4 3 = log 4 3 – log 47 Quotient property 7 Use the given values of log 4 3 and 0. 792 – 1. 404 log 7. 4 = – 0. 612 b. log 4 21 = log 4 (3 7) Simplify. Write 21 as 3 7. = log 43 + log 47 Product property 0. 792 + 1. 404 Use the given values of log 4 3 and log 47. = 2. 196 Simplify.

Warm-Up 1 Exercises EXAMPLE Use properties of logarithms Use log 4 3 0. 792 and log 47 logarithm. c. log 4 49 1. 404 to evaluate the = log 4 72 Write 49 as 72 = 2 log 4 7 Power property 2(1. 404) = 2. 808 Use the given value of log 47. Simplify.

Warm-Up YOU TRY Exercises for Example 1 Use log 6 5 0. 898 and log 6 8 1. 161 to evaluate the logarithm. 1. log 6 5 = log 6 5 – log 68 Quotient property 8 Use the given values of log 6 5 and 0. 898 – 1. 161 log 8. 6 = – 0. 263 2. log 6 40 = log 6 (8 5) Simplify. Write 40 as 8 5. = log 68 + log 65 Product property 1. 161 + 0. 898 Use the given values of log 6 5 and log 68. = 2. 059 Simplify.

Warm-Up YOU TRY Exercises for Example 1 Use log 6 5 0. 898 and log 6 8 logarithm. 3. log 6 64 1. 161 to evaluate the = log 6 82 Write 64 as 82 = 2 log 6 8 Power property 2(1. 161) = 2. 322 4. log 6 125 = log 6 53 = 3 log 6 5 3(0. 898) = 2. 694 Use the given value of log 68. Simplify. Write 125 as 53 Power property Use the given value of log 65. Simplify.

Warm-Up 2 Exercises EXAMPLE Expand a logarithmic expression Expand log 6 5 x 3 y SOLUTION log 6 5 x 3 = log 6 5 x 3 – log 6 y y = log 6 5 + log 6 x 3 – log 6 y = log 6 5 + 3 log 6 x – log 6 y Quotient property Product property Power property

Warm-Up 3 Exercises EXAMPLE Condensing Logarithmic Expressions SOLUTION log 9 + 3 log 2 – log 3 = log 9 + log 23 – log 3 Power property = log (9 23) – log 3 3 9 2 = log 3 = log 24 ANSWER The correct answer is D. Product property Quotient property Simplify.

Warm-Up YOU TRY Exercises for Examples 2 and 3 5. Expand log 3 x 4. SOLUTION log 3 x 4 = log 3 + log x 4 = log 3 + 4 log x Product property Power property

Warm-Up YOU TRY Exercises for Examples 2 and 3 6. Condense ln 4 + 3 ln 3 – ln 12. SOLUTION ln 4 + 3 ln 3 – ln 12 = ln 4 + ln 33 – ln 12 = ln (4 33) – ln 12 3 4 3 ln = 12 = ln 9 Power property Product property Quotient property Simplify.

Warm-Up Exercises Warm-Up Use log 5 20 logarithm. 1. log 5 160 ANSWER 2. 1. 861 and log 5 8 3. 153 log 5 8000 ANSWER 5. 583 1. 292 to evaluate the

Warm-Up Exercises Warm-Up Use log 5 20 logarithm. 3. Expand In ANSWER 4. 1. 861 and log 5 8 3 y 1. 292 to evaluate the . 23 1 In x – 2 In y 3 Condense 5 log 2 x – 4 log 2 y. ANSWER 5 log 2 x 4. y

Warm-Up Exercises

Warm-Up 4 Exercises EXAMPLE Use the change-of-base formula Evaluate log 3 8 using common logarithms and natural logarithms. SOLUTION Using common logarithms: log 8 0. 9031 log 3 8 = 0. 4771 log 3 Using natural logarithms: ln 8 2. 0794 log 3 8 = 1. 0986 ln 3 1. 893

Warm-Up 5 Exercises EXAMPLE Use properties of logarithms in real life Sound Intensity 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 L(I) = 10 log I I 0 where I 0 is the intensity of a barely audible sound (about 10– 12 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?

Warm-Up 5 Exercises EXAMPLE Use properties of logarithms in real life SOLUTION Let I be the original intensity, so that 2 I is the doubled intensity. Increase in loudness = L(2 I) – L(I) 2 I – 10 log I = 10 log I 0 2 I – log I = 10 log I 0 = 10 log 2 + log I – log I I 0 = 10 log 2 3. 01 ANSWER Write an expression. Substitute. Distributive property Product property Simplify. Use a calculator. The loudness increases by about 3 decibels.

Warm-Up YOU TRY Exercises for Examples 4 and 5 Use the change-of-base formula to evaluate the logarithm. 7. log 5 8 SOLUTION log 8 log 5 8 = log 5 0. 9031 0. 6989 1. 292 8. log 8 14 SOLUTION log 14 log 8 14 = log 8 1. 146 0. 9031 1. 269

Warm-Up YOU TRY Exercises for Examples 4 and 5 Use the change-of-base formula to evaluate the logarithm. 9. log 26 9 SOLUTION log 9 log 26 9 = log 26 0. 9542 1. 4149 0. 674 1. 4777 1. 076 1. 369 10. log 12 30 SOLUTION log 12 30 = log 30 log 12

Warm-Up YOU TRY Exercises 11. for Examples 4 and 5 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? SOLUTION L(I) = 10 log I I 0 Let I be the original intensity, so that 3 I is the tripled intensity.

Warm-Up YOU TRY Exercises for Examples 4 and 5 Increase in loudness = L(3 I) – L(I) 3 I – 10 log I = 10 log I 0 3 I – log I = 10 log I 0 = 10 log 3 + log I – log I I 0 = 10 log 3 4. 771 Write an expression. Substitute. Distributive property Product property Simplify. Use a calculator. ANSWER The loudness increases by about 4. 771 decibels.

Warm-Up Exercises KEEP GOING Use log 5 20 logarithm. 1. 861 and log 5 8 1. 292 to evaluate the 1. Use the change-of- base formula to evaluate log 4 50. ANSWER 2. 822 2. The intensity level of an electric guitar is 102. 8 watts per square meter. Use the formula L (I)=10 log II 0 where I 0 10– 12 watts per square meter, to find the decibel level of the guitar. ANSWER about 28 decibels