2 logb (MN) = logb M + logb N The Product RuleLet b, M, and N be positive real numbers with b 1.logb (MN) = logb M + logb NThe logarithm of a product is the sum of the logarithms.For example, we can use the product rule to expand ln (4x):ln (4x) = ln 4 + ln x.
3 The Quotient Rule log M N æ è ç ö ø ÷ = - Let b, M and N be positive real numbers with b 1.The logarithm of a quotient is the difference of the logarithms.logbMNæèçöø÷=-
4 The Power RuleLet b, M, and N be positive real numbers with b = 1, and let p be any real number.log b M p = p log b MThe logarithm of a number with an exponent is the product of the exponent and the logarithm of that number.
5 Text Example Write as a single logarithm: a. log4 2 + log4 32 Solution a. log4 2 + log4 32 = log4 (2 • 32) Use the product rule.= log4 64= 3Although we have a single logarithm, we can simplify since 43 = 64.
6 ProblemsWrite the following as single logarithms:
7 The Change-of-Base Property For any logarithmic bases a and b, and any positive number M,The logarithm of M with base b is equal to the logarithm of M with any new base divided by the logarithm of b with that new base.
8 Example: Changing Base to Common Logs Use common logarithms to evaluate log5 140.Solution BecauseThis means thatExample: Changing Base to Natural LogsUse natural logarithms to evaluate log5 140.Solution BecauseThis means that
9 ExampleUse logarithms to evaluate log37.Solution:orso