Exponential and Logarithmic Functions

Properties of Logarithms

This section explores the fundamental properties of logarithms, which are essential tools for simplifying and manipulating logarithmic expressions in algebra. Mastery of these properties allows for the expansion, condensation, and evaluation of logarithmic expressions.

Objectives

• Use the product rule for logarithms

• Use the quotient rule for logarithms

• Use the power rule for logarithms

• Expand logarithmic expressions

• Condense logarithmic expressions

• Use the change-of-base property

The Product Rule

The product rule for logarithms states that the logarithm of a product is equal to the sum of the logarithms of the factors. For any positive real numbers b, M, and N with b \ne 1:

• Key Point: This property allows you to break down the logarithm of a product into the sum of two simpler logarithms.

• Example: Expand using the product rule:

The Quotient Rule

The quotient rule for logarithms states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator. For any positive real numbers b, M, and N with b \ne 1:

• Key Point: This property allows you to express the logarithm of a quotient as the difference of two logarithms.

• Example: Expand :

The Power Rule

The power rule for logarithms states that the logarithm of a number raised to an exponent is equal to the exponent times the logarithm of the base number. For any positive real numbers b, M with b \ne 1, and any real number p:

• Key Point: This property allows you to move the exponent in a logarithmic expression to the front as a multiplier.

• Example: Expand :

Expanding Logarithmic Expressions

To expand logarithmic expressions, apply the product, quotient, and power rules to write a single logarithm as a sum, difference, or multiple of simpler logarithms.

• Example: Expand : Step 1: Apply the quotient rule: Step 2: Apply the product rule to the numerator: Step 3: Apply the power rule:

Condensing Logarithmic Expressions

To condense logarithmic expressions, combine sums, differences, and multiples of logarithms into a single logarithm using the product, quotient, and power rules in reverse.

• Example: Condense : Step 1: Apply the power rule in reverse: Step 2: Apply the product rule: Step 3: Apply the quotient rule:

The Change-of-Base Property

The change-of-base property allows you to rewrite a logarithm in terms of logarithms with a different base. For any positive numbers a, b, and M with a \ne 1 and b \ne 1:

• Key Point: This property is especially useful for evaluating logarithms on calculators, which typically only have keys for common logarithms (base 10) and natural logarithms (base e).

• Example: Evaluate using common logarithms:

Common and Natural Logarithms

• Common logarithms are logarithms with base 10, written as or simply .

• Natural logarithms are logarithms with base e (where ), written as or .

• Calculators typically provide buttons for and .

• Example: Evaluate using natural logarithms: