Logarithmic Functions Lesson 8.4

Vocabulary Common Logarithm: the logarithm with base 10. It is denoted by log10 or simply by log. Natural Logarithm: the logarithm with base e. It can be denoted by loge but it is more often denoted by ln.

Definition of Logarithm with Base b Let b and y be positive numbers , b ≠ 1. The logarithm of y with base b is denoted by logb y and is defined as follows: logb y = x if and only if bx = y The expression logb y is read as “log base b of y”.

Example 1: Rewriting Logarithmic Equations Logarithmic Form log3 81 = 4 log4 1 = 0 log9 9 = 1 log log3 3 = 1 log2 .125 = -3 Exponential Form

Special Logarithmic Values Let b be a positive real number such that b ≠ 1. Logarithm of 1 : logb 1 = 0 because b0 = 1 Logarithm of base b : logb b = 1 because b1 = b

Example 2: Evaluating Logarithmic Expressions

Example 3: Using Inverse Properties 10log 2.3 Log2 8x 10log x Log3 81x

Example 4: Finding Inverses y = log y = ln (x – 2) y = log2 x

Graphs of Logarithmic Functions The graph of y = logb (x – h) + k has these characteristics: The line x = h is a vertical asymptote. The domain is x > h, and the range is all real numbers If b > 1 the graph moves up to the right. If 0<b<1, the graph moves down to the right.

Example 5: Graphing Logarithmic Functions A) y = log B) y = log2 (x + 1) + 2