4.3 Logarithm Functions Recall: a ≠ 1 for the exponential function f(x) = a x, it is one-to-one with domain (-∞, ∞) and range (0, ∞). when a > 1, it is always increasing when 0 < a < 1, it is always decreasing In both cases, the exponential function has an inverse which is called the Logarithm Function with base a. The Logarithm Function with Base a For each positive number a ≠ 1 and each x in (0, ∞), and for each x in (0, ∞) and each real number y.

Ex 1: Use the inverse relationship with the exponential functions to determine x in the following. Solution: a.) From the definition, the conversion implies that (x – 4)(x + 2), so x = 4 or x = 2

Note: The logarithm function is the inverse of the exponential function. It is the graph of the exponential function reflected over the line y = x. Graph on board: Note: The inverses of the exponential functions intersect at (1, 0) and x = 0 is a vertical asymptote. Ex 2: Sketch the graphs of the following. is the inverse of the function is still the inverse of the function We shift one unit right then 3 units up.

Arithmetic Properties of Logarithms: For each positive number a ≠ 1, each pair of positive real numbers x 1, x 2 and each real number r we have, Note: Just as exponential functions have inverses, so does the natural exponential function. (e).

The Natural Logarithm Function For each x in (0, ∞), we have, for each x in (0, ∞) and each real number y. Arithmetic Properties of Natural Logarithms For each positive number a ≠ 1, each pair of positive real numbers x 1, x 2, and each real number r we have,

Ex 3: Determine the values of x that satisfy each expresssion. Solution: a.) The product rule implies that… Taking the inverse of both sides (e) cancels out the “ln” leaving… Simplify…1 = x – 2 so, x = 3

b.) Here, we use the product and exponent rules ln 3 + ln (2x – 1) = ln 3(2x – 1) = ln (6x – 3) and… So we now have… Taking the inverse of both sides…

c.) We can rewrite the equation as… Using the arithmetic properties… Changing this to its equivalent exponential form… So… x = 5 or x = -1

d.) We can rewrite this equation as… Multiply both sides by 2 Take inverse of both sides Solve for x You could verify that this is a solution by substituting in back into the original; you will see it equals ½.