Properties of Logarithms

The Product Rule Let b, M, and N be positive real numbers with b  1. logb (MN) = logb M + logb N The 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.

The Quotient Rule Let b, M and N be positive real numbers with b  1. The logarithm of a quotient is the difference of the logarithms.

The Power Rule Let 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 M The logarithm of a number with an exponent is the product of the exponent and the logarithm of that number.

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 = 3 Although we have a single logarithm, we can simplify since 43 = 64.

Properties for Expanding Logarithmic Expressions For M > 0 and N > 0:

Example Use logarithmic properties to expand the expression as much as possible.

Example cont.

Example cont.

Properties for Condensing Logarithmic Expressions For M > 0 and N > 0:

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.

Example Use logarithms to evaluate log37. Solution: or so

Properties of Logarithms