Copyright © 2011 Pearson Education, Inc. Rules of Logarithms Section 4.3 Exponential and Logarithmic Functions

4.3 Copyright © 2011 Pearson Education, Inc. Slide 4-3 Inverse Rules If a > 0 and a ≠ 1, then 1. log a (a x ) = x for any real number x, and 2. The Inverse Rules

4.3 Copyright © 2011 Pearson Education, Inc. Slide 4-4 Product Rule for Logarithms For M > 0 and N > 0, log a (MN) = log a (M) + log a (N). The Logarithm of a Product

4.3 Copyright © 2011 Pearson Education, Inc. Slide 4-5 PROOF Using the product rule for exponents and the inverse rule, Now by the definition of logarithm we have The Logarithm of a Product

4.3 Copyright © 2011 Pearson Education, Inc. Slide 4-6 Quotient Rule for Logarithms For M > 0 and N > 0, Power Rule for Logarithms For M > 0 and any real number N, log a (M N ) = N · log a (M). The Logarithm of a Quotient and the Logarithm of a Power

4.3 Copyright © 2011 Pearson Education, Inc. Slide 4-7 Rules of Logarithms with Base a If M, N, and a are positive real numbers with a ≠ 1, and x is any real number, then 1. log a (a) = 12. log a (1) = 0 3. log a (a x ) = x4. 5. log a (MN) = log a (M) + log a (N) 6. log a (M/N) = log a (M) – log a (N) 7. log a (M x ) = x · log a (M)8. log a (1/N) = – log a (N) Using the Rules

4.3 Copyright © 2011 Pearson Education, Inc. Slide 4-8 Rules of Natural Logarithms If M and N are positive real numbers and x is any real number, then 1. ln( e ) = 1 2. ln(1) = 0 3. ln( e x ) = x ln(MN) = ln(M) + ln(N) 6. ln(M/N) = ln(M) – ln(N) 7. ln(M x ) = x · ln(M) 8. ln(1/N) = – ln(N) Using the Rules

4.3 Copyright © 2011 Pearson Education, Inc. Slide 4-9 Base-Change Formula If a > 0, b > 0, a ≠ 1, b ≠ 1, and M > 0, then The Base-Change Formula

4.3 Copyright © 2011 Pearson Education, Inc. Slide 4-10 The inverse rules are easy to use if you remember that log a (x) is the power of a that produces x. The product rule for logarithms says that the logarithm of a product of two numbers is equal to the sum of their logarithms, provided that all of the logarithms are defined and all have the same base. The quotient rule for logarithms says that the logarithm of a quotient of two numbers is equal to the difference of their logarithms, provided that all logarithms are defined and all have the same base. The power rule for logarithms says that the logarithm of a power of a number is equal to the power times the logarithm of the number, provided that all logarithms are defined and have the same base. The base-change formula says that the logarithm of a number in one base is equal to the logarithm of the number in the new base divided by the logarithm of the old base. Review of Rules