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College Algebra Chapter 4 Exponential and Logarithmic Functions Section 4.4 Properties of Logarithms Concepts 1. Apply the Product, Quotient, and Power Properties of Logarithms 2. Write a Logarithmic Expression in Expanded Form 3. Write a Logarithmic Expression as a Single Logarithm 4. Apply the Change-of-Base Formula Apply the Product, Quotient, and Power Properties of Logarithms Let b, x, and y be positive real numbers where b ≠ 1. Product Property: logb ( xy) logb x logb y Quotient Property: x log b log b x log b y y Power Property: log b x p log b x p For these exercises, assume that all variable expressions represent positive real numbers. Examples 1 – 3: Use the product property of logarithms to write the logarithm as a sum. Then simplify if possible. 1. log 2xy 3. ln 3(a b) 2. log 2 2xy Examples 4 – 6: Use the quotient property of logarithms to write the logarithm as a difference. Then simplify if possible. 4. z log 7 49 6. 100 log x y 5. a log 7 14 Examples 7 – 9: Apply the power property of logarithms. 7. 9. ln x 3 5 log x 2 8. ln e 5 Concepts 1. Apply the Product, Quotient, and Power Properties of Logarithms 2. Write a Logarithmic Expression in Expanded Form 3. Write a Logarithmic Expression as a Single Logarithm 4. Apply the Change-of-Base Formula Example 10: Write the expression as the sum or difference of logarithms. 5z log 3 w Example 11: Write the expression as the sum or difference of logarithms. ac log 7 5d Example 12: Write the expression as the sum or difference of logarithms. 2 ln x y Example 13: Write the expression as the sum or difference of logarithms. 3 x2 ln w z Example 14: Write the expression as the sum or difference of logarithms. log 2 4x yz 3 Example 15: Write the expression as the sum or difference of logarithms. 64 x 2 y log8 3 3zw Concepts 1. Apply the Product, Quotient, and Power Properties of Logarithms 2. Write a Logarithmic Expression in Expanded Form 3. Write a Logarithmic Expression as a Single Logarithm 4. Apply the Change-of-Base Formula Example 16: Write the logarithmic expression as a single logarithm with a coefficient of 1, and simplify as much as possible. log( 4 x 3) log x Example 17: Write the logarithmic expression as a single logarithm with a coefficient of 1, and simplify as much as possible. log 2 z 4 log 2 y Example 18: Write the logarithmic expression as a single logarithm with a coefficient of 1, and simplify as much as possible. 3 ln x ln( x 2) ln 5 Example 19: Write the logarithmic expression as a single logarithm with a coefficient of 1, and simplify as much as possible. 2 2 2 ln x ln x ln z 9 ln z 3 Example 20: Write the logarithmic expression as a single logarithm with a coefficient of 1, and simplify as much as possible. 2 log 5 x 5 log5 x log5 x 2 25 Examples 21 – 23: Use logb 2 0.4307, logb 3 0.6826, and logb 7 1.2091 to approximate the value of 21. logb 21 23. 7 log b 2 22. logb 9 Concepts 1. Apply the Product, Quotient, and Power Properties of Logarithms 2. Write a Logarithmic Expression in Expanded Form 3. Write a Logarithmic Expression as a Single Logarithm 4. Apply the Change-of-Base Formula Apply the Change-of-Base Formula Let a and b be positive real numbers such that a ≠ 1 and b ≠ 1. Then for any positive real number x log b x log a x log a b In particular, log b x log x log b or ln x ln b Examples 24, 25: Use the change-of-base formula and a calculator to approximate the logarithm to 4 decimal places. 24. log 7 15 25. log5 0.3