Chapter 4 – Exponential and Logarithmic Functions 4.3 - Logarithmic Functions.

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Presentation transcript:

Chapter 4 – Exponential and Logarithmic Functions Logarithmic Functions

Exponential Functions Recall from last class that every exponential function f (x) = a x with a >0 and a  1 is a one- to-one function and therefore has an inverse function. That inverse function is called the logarithmic function with base a and is denoted by log a Logarithmic Functions

Definition Logarithmic Function Let a be a positive number with a  1. The logarithmic function with base a, denoted by log a, is defined by log a x = y  a y = x So log a x is the exponent to which the base a must be raised to give x Logarithmic Functions

Switching Between Logs & Exp. NOTE: log a x is an exponent! Logarithmic Functions

Example – pg. 322 #5 Complete the table by expressing the logarithmic equation in exponential form or by expressing the exponential equation into logarithmic form Logarithmic Functions

Properties of Logarithms Logarithmic Functions

Example – pg. 322 Use the definition of the logarithmic function to find x Logarithmic Functions

Graphs of Logarithmic Functions Because the exponential and logarithmic functions are inverses with each other, we can learn about the logarithmic function from the exponential function. Remember, Logarithmic Functions CharacteristicExponentialLogarithmic Domain(-∞, ∞) Range(0, ∞) x-interceptNone y-intercept(0,1) VANone HAy = 0

Graphs of Log Functions Logarithmic Functions

Example – pg. 323 Graph the function, not by plotting points or using a graphing calculator, but by starting from the graph of a log a x function. State the domain, range, and asymptote Logarithmic Functions

Definitions Common Logarithm The logarithm with base 10 is called the common logarithm and is denoted by omitting the base: Natural Logarithm The logarithm with base e is called the natural logarithm and is denoted by: Logarithmic Functions

Note Both the common and natural logs can be evaluated on your calculator Logarithmic Functions

Properties of Natural Logs Logarithmic Functions

Example – pg. 322 Find the domain of the function Logarithmic Functions