Logarithmic Functions & Their Graphs 3.2

Inverses of Exponentials Remember from section 1.6 that if a function is one-to-one (meaning it passes the Horizontal Line Test) it must have an inverse function. Looking at the graphs of exponential functions of the form f(x) = ax, they all pass the Horizontal Line Test and therefore have an inverse function. The inverse function is called the logarithmic function with base a.

Definition of Logarithmic Functions For x > 0, a > 0, and a ≠ 1 y = loga x If and only if x = ay. The function given by f(x) = loga x is called the logarithmic function with base a. Read as log base a of x.

When evaluating logarithms, remember that loga x is the exponent to which a must be raised to obtain x. For example: log2 8 = 3 because 23 = 8.

Example 1: Evaluating Logarithms Use the definition of logarithmic functions to evaluate each logarithm at the indicated value of x. f(x) = log2 x x = 32 F(x) = log3 x x = 1 F(x) = log4 x x = 2 F(x) = log10 x x =

Common Logarithm The logarithmic function with base 10 is called the common logarithmic function.

Example 2: Evaluating Common Logarithms on a Calculator Use a calculator to evaluate the function f(x) = log10 x at each value of x. x = 10 x = 2.5 x = -2 x =

Error Message In part c we obtain an error message because the domain of every logarithmic function is the set of positive real numbers.

Properties of Logarithms loga 1 = 0 because a0 = 1 loga a = 1 because a1 = a loga ax = x and alogx = x INVERSES If loga x = loga y then x = y

Example 3: Using Properties of Logarithms Solve for x: log2 x = log2 3 Solve for x: log4 4 = x Simplify: log5 5x Simplify: 7log7 14

The Natural Logarithmic Function The function f(x) = ex is one-to-one and has an inverse function. This inverse function is called the natural logarithmic function and is denoted by ln x (read as the natural log of x).

For x > 0, y = ln x if and only if x = ey For x > 0, y = ln x if and only if x = ey. The function given by f(x) = loge x = ln x is called the natural logarithmic function. NOTE: the natural logarithm ln x is written without a base. The base is understood to be e.

Example 4: Evaluating the Natural Logarithmic Function Use a calculator to evaluate the function f(x) = ln x at each indicated value of x. x = 2 x = .3 x = -1

Properties of Natural Logarithms ln 1 = 0 because e0 = 1 ln e = 1 because e1 = e ln ex = x and elnx = x INVERSES If ln x = ln y, then x = y

Example 5: Using Properties of Natural Logarithms Use the properties of natural logarithms to rewrite each expression. ln eln 5 ln e0 2 ln e

Example 6: Finding the Domains of Logarithmic Functions Find the domain of each function. f(x) = ln (x – 2) g(x) = ln (2 – x) h(x) = ln x2