Properties of Logarithmic Functions Section 5.4 Properties of Logarithmic Functions Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.

Objectives Convert from logarithms of products, powers, and quotients to expressions in terms of individual logarithms, and conversely. Simplify expressions of the type logaax and

Logarithms of Products The Product Rule For any positive numbers M and N and any logarithmic base a, loga MN = loga M + loga N. (The logarithm of a product is the sum of the logarithms of the factors.)

Example Express as a single logarithm:

Logarithms of Powers The Power Rule For any positive number M, any logarithmic base a, and any real number p, (The logarithm of a power of M is the exponent times the logarithm of M.)

Example Express as a product.

Logarithms of Quotients The Quotient Rule For any positive numbers M and N, and any logarithmic base a, (The logarithm of a quotient is the logarithm of the numerator minus the logarithm of the denominator.)

Example Express as a difference of logarithms:

Example Express as a single logarithm:

Example Express each of the following in terms of sums and differences of logarithms.

Example (continued)

Example (continued)

Example Express as a single logarithm:

Example Given that loga 2 ≈ 0.301 and loga 3 ≈ 0.477, find each of the following, if possible.

Examples (continued) Cannot be found using these properties and the given information.

Expressions of the Type loga ax The Logarithm of a Base to a Power For any base a and any real number x, loga ax = x. (The logarithm, base a, of a to a power is the power.)

Examples Simplify. a) loga a8 b) ln et c) log 103k a. loga a8

Expressions of the Type A Base to a Logarithmic Power For any base a and any positive real number x, (The number a raised to the power loga x is x.)

Example Simplify.