 | Logarithmic Functions MathBitsNotebook.com Topical Outline | Algebra 2 Outline | MathBits' Teacher Resources Terms of Use Contact Person: Donna Roberts  | Let's start by taking a look at the inverse of the exponential function, f (x) = 2x . In the lesson "Intro to Inverses of Functions", we saw that the inverse of a function is found when the inputs (x) and outputs (y) exchanged places. The graph of the inverse is a reflection of the original function over the identity line y = x. f (x) = 2x | x | -2 | -1 | 0 | 1 | 2 | 3 | | f(x) | ¼ | ½ | 1 | 2 | 4 | 8 | The inverse: f -1 (x) | x | ¼ | ½ | 1 | 2 | 4 | 8 | | f-1(x) | -2 | -1 | 0 | 1 | 2 | 3 | This new inverse function is called a logarithmic function and is expressed by the equation: y = log2 (x) |  For f (x): Domain (-∞,∞), Range (0,∞) For g (x): Domain (0,∞), Range (-∞,∞) | 
 Logs are Exponents! | The logarithmic function is the function  where b is any number such that b > 0, b≠ 1, and x > 0. The function is read "log base b of x". The logarithm y is the exponent to which b must be raised to get x. |  The inverse of the exponential function y = bx, will be x = by (where the x and y change places). Note that in the inverse, the y (the logarithm) becomes an exponent. | | | Let's look more closely at the graph: y = log2 x The value of b (the 2) is referred to as the base of the logarithm. Notice that x must be positive. The parent function is f (x) = logb x | |  | Most logarithmic graphs resemble this same basic shape. Notice that this graph is very, very close to the y-axis but it does not intersect with, nor cross, the y-axis. The x-values of this graph are always positive, and the y-values increase as the graph progresses to the right (as seen in the above graph). This graph has a vertical asymptote at x = 0. Note: In a logarithmic graph, the "rate of change" increases (or decreases) across the graph. | Properties of Logarithmic Functions: f (x) = logb x | | The function f (x) = logb x features: • a domain of positive real numbers, never zero (0,∞). • a range of all real numbers (-∞,∞). • an x-intercept at (1,0). • a vertical asymptote at x = 0 (the y-axis). • an increasing graph when b > 1. • a decreasing graph when 0 < b < 1. • a graph that passes the vertical line test for functions. • a graph that passes the horizontal line test for functional inverse. • a graph that displays a one-to-one function. | | Parameter: b - the base, controls the rate of change of the function |  y = log(x) implies y = log10(x) called the common log (base 10).
| | A few pointers about "intercepts" in logarithmic functions: | | | y = logb x | y = a logb x  | By examining the nature of the logarithmic graph, we have seen that the parent function will stay to the right of the x-axis, unless acted upon by a transformation. The parent function, , will always have an x-intercept of one, occurring at the ordered pair of (1,0). . There is no y-intercept with the parent function since it is asymptotic to the y-axis (approaches the y-axis but does not touch or cross it). | The transformed parent function of the form y = a logb x, will also always have a x-intercept of 1, occurring at the ordered pair of (1, 0). Note that the value of a may be positive or negative. Like the parent function, this transformation will be asymptotic to the y-axis, and will have no y-intercept. | | B U T . . . Transformations can change the game!If a transformed logarithmic function includes, for example, a vertical or horizontal shift, all bets are off. The horizontal shift can affect the x-intercept and the possibility of a y-intercept, while the vertical shift can affect the x-intercept. In these situations, you will need to examine the graph carefully to determine what is happening. |  | | | The End Behavior of Logarithmic Functions | | The end behavior of a logarithmic graph also depends upon whether you are dealing with the parent function or with one of its transformations. | • The end behavior of the parent function is consistent. As x approaches infinity, the y-values slowly get larger, approaching infinity. As x approaches 0 from the right (denoted as x→ 0+), the y-values approach negative infinity. |  | Find important graphing information at:  | How to use your graphing calculator for working with logarithms Click here. | | |  | How to use your graphing calculator for working with logarithms, Click here. | | | | NOTE: The re-posting of materials (in part or whole) from this site to the Internet is copyright violation and is not considered "fair use" for educators. Please read the "Terms of Use". | Topical Outline | Algebra 2 Outline | MathBitsNotebook.com | MathBits' Teacher Resources Terms of Use Contact Person: Donna Roberts |
The y-intercept, where x = 0, is