9.3 Logarithmic Functions

To solve a logarithmic equation, it is often best to start by changing it to its exponential equivalent. Ex 1) Solve for x a)log 7 x = = x 49 = x f (x) = b x is one-to-one so it has an inverse. The inverse of an exponential function is called a logarithmic function. For positive real numbers x and b, b > 0 and b ≠ 1, D = x > 0 R =  D =  R = y > 0 inverses Do “around the world” start with base c) log x 81 = 4 x 4 = 81 x = 3 b)

Ex 2) Graph. Find Domain, Range, x- & y-int, asymptote, inc or dec. a) f (x) = log 2 x xy –1 ¼ 2 –1 = = –2 2 –2 = = 2 2 = 4 ½ 2 xy – –2 2 1 ¼ 2 ½ b) D: x > 0, R: , x-int: (1, 0), y-int: none, increasingasympt: x = 0, *Note: For f (x) = log b x if b > 1, f (x) increases if 0 < b < 1, f (x) decreases!  y = log 2 x 2 y = x Plug values into y D: x > 0, R: , x-int: (1, 0), y-int: none, decreasingasympt: x = 0,

The base of a log function can be any positive number except 1. But, there are two popular & powerful common bases. These have MANY applications to science & engineering (we’ll see tomorrow) Basic Log Facts: (common log)(natural log)and (written as log x)(written as ln x)

Ex 3) Simplify each expression. a)log (log fact!) b)log 3 81 log c)ln e 3 log e e 3 3 d) log –2 –2 e) undefined why??? f) e ln6 e loge6 6

Homework # Logarithmic Functions WS