 | 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 Algebra 1, you saw that when working with the inverse of a function, the inputs (x) and outputs (y) exchange places, and that the inverse will be a reflection 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) |  | The composition of a function with its inverse returns the starting value, x.  This concept will be used to solve equations involving exponentials and logarithms.  Now that we have a basic idea of a logarithmic function, let's take a closer look at its graph.
 | 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 y = bx, will be x = by (where the x and y change places). Note that y (the logarithm) is actually an exponent. | | | Let's examine the function:  The value of b (the 2) is referred to as the base of the logarithm. Notice that x must be positive. | |  | Most logarithmic graphs resemble this same basic shape. Notice that this graph is very, very close to the y-axis but does not cross it. 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). Note: In a linear graph, the "rate of change" remains the same across the entire graph. In a logarithmic graph, the "rate of change" increases (or decreases) across the graph. | Common Logarithm Base 10 logarithm log10(x) abbreviated log(x) | | | Natural Logarithm Base e logarithm loge(x) abbreviated ln(x) | | | Characteristics of Logarithmic Functions | | The graphs of functions of the form  have certain characteristics in common.  Logarithmic functions are one-to-one functions. | • graph crosses the x-axis at (1,0) • when b > 1, the graph increases • when 0 < b < 1, the graph decreases • the domain is all positive real numbers (never zero) • the range is all real numbers • graph passes the vertical line test for functions • graph passes the horizontal line test for functional inverse. • graph is asymptotic to the y-axis - gets very, very close to the y-axis but, in this case, does not touch it or cross it. | | Transformations on Logarithmic Functions | | We know that transformations have the ability to move functions by sliding them, reflecting them, stretching them, and shrinking them. Let's see how these changes will affect the logarithmic function: Parent function: y = a logbx Stretch (|a| > 1): Compress or Shrink (0 < |a| < 1):  Domain: x > 0 Range: x ∈ Real numbers | y = a logbx Reflection (a < 0) in x-axis: Domain: x > 0 Range: x ∈ Real numbers | Translation y = logb(x - h) + k horizontal by h: vertical by k:  Domain: x > h Range: x ∈ Real numbers
| All 3 transformations combined: y = a logb(x - h) + k | Intercepts of Logarithmic Functions | | 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, y = logb x, 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. | • If the transformed parent function includes a vertical or horizontal shift, all bets are off. The horizontal shift will affect the possibility of a y-intercept and the vertical shift will affect the x-intercept. In this situation, you will need to examine the graph carefully to determine what is happening. |  | | 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. |  | | 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 |
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