5.4 Logarithmic Functions and Models Evaluate the common logarithmic function Evaluate logarithms with other bases Solve basic exponential and logarithmic equations Solve general exponential and logarithmic equations Convert between exponential and logarithmic forms

Common Logarithm (1 of 3)

Common Logarithm (2 of 3)

Common Logarithm (3 of 3) The common logarithm of a positive number x, denoted log x, is defined by where k is a real number. The function given by f(x) = log x is called the common logarithmic function.

Evaluate

Example: Evaluating common logarithms Simplify each logarithm by hand.

Graphs of f(x) = 10x and f –1(x) = log x Their graphs are reflections across the line y = x.

Inverse Properties of the Common Logarithm The following inverse properties hold for the common logarithm.

Base-2 Logarithm x 2−3.1 2−2 2−0.5 20 20.5 22 23.1 log2x −3.1 −2 −0.5 0.5 2 3.1 A base-2 logarithm is an exponent of 2.

Base-e Logarithm - Natural Logarithm x e−3.1 e−2 e−0.5 e0 e0.5 e2 e3.1 ln x −3.1 −2 −0.5 0.5 2 3.1 A natural logarithm is an exponent of e.

Logarithm

Example: Evaluating logarithms Evaluate each logarithm.

Inverse Properties The following inverse properties hold for logarithms with base a.

Example: Applying inverse properties (1 of 2) Use inverse properties to evaluate each expression.

Example: Applying inverse properties (2 of 2)

Example: Solving a base-10 exponential equation

Example: Solving a common logarithmic equation

Example: Solving exponential equations

Example: Solving logarithmic equations