Finding Domain and Range of Logarithmic Functions

Description/Explanation/Highlights

Video Description

This video explains how to find the domain and range of the logarithmic parent and other logarithmic functions.

Steps and Key Points to Remember

To find the domain and range of logarithmic functions, follow these steps:

  1. Logarithmic functions have multiple parent functions; one for each base, but all logarithmic parents have a couple of things in common They all have a vertical asymptote (a vertical line the graph approaches but never reaches) at x = 0 (the y-axis) and the graph will intersect the x-axis at point (1, 0).
  2. To find the domain for any parent function such as \(y=\log_2(x)\) (the parent for a base 2 log), simply realize that there is a vertical asymptote at x = 0 which will prevent the graph from ever touching or reaching the y-axis but all x-values to the right are defined. This tells us that the domain (possible x-values) is greater than 0.
  3. The domain can be written as \(x>0\) or as \(0<x<\infty\) or in interval notation as \((0,\infty)\). Since all logarithmic parent functions have a vertical asymptote at x = 0, the domain for all parent functions in any base will be the same.
  4. Since the value of a Log can be positive, negative or zero (remember logs represent exponents), the range of all logarithmic functions will be all real numbers. This can be written as \(\mathbb{R} \) or as \((-\infty<x<\infty)\) or in interval notation as \((-\infty, \infty)\).
  5. A transformation of the log parent such a \(y=\log_2(x) +4\) causes the entire parent graph to be moved up 4 units (-4 would have been down). Since the graph only moves up (or down), the asymptote is still at x = 0 so the domain doesn’t change from the parent. So in an up or down translation, the domain remains: \(0<x<\infty\) and the range is always all real numbers.
  6. Putting the transformation “inside” the function such as \(y=\log_2(x+4)\) does have an effect on the domain of the function. Since adding 4 “inside” the function moves the graph AND the vertical asymptote 4 places left (- moves right), the domain is changed. Now the x-values approach -4 instead of 0, so the new domain is \(-4<x<\infty\) and as always, the range remains all real numbers.
  7. When there is a number in front of the variable on the “inside” of the function as in the function \(y=\log_2(4x+12)\), first divide all terms inside the function by the coefficient in front of x. The number that is added (or subtracted) now tells you how far left (+) or right (-) to move the asymptote and thus where the domain begins.
  8. In this example, dividing by 4 results in +3 instead of 12 and tells us that the new asymptote is at -3 (3 units left). Therefore the domain is \(-3<x<\infty\).
  9. When a number is in front of the function but on the outside such as the -2 in the function \(y=-2\log(2x-10)+4\), the graph is stretched vertically but is not moved from side to side and doesn’t affect the domain. The only transformation that has an effect on the domain is a horizontal translation caused by numbers added or subtracted inside the function. In this function, divide the two terms inside the function by the 2 in front of x leaving -5 as the number subtracted. This moves the asymptote to the right 5 units and means the domain now starts at 5, so the domain is \(5<x<\infty\).

Here are some key points to keep in mind when finding the domain and range of logarithmic functions.

  • All logarithmic parent functions take the form of \(y=\log_b(x)\) and have a vertical asymptote at x = 0 (the y-axis) and the graph crosses x at point (1, 0).
  • The left boundary of the domain is controlled by the vertical asymptote.
  • The right boundary is always infinity.
  • The domain of all parent functions is always: \(0<x<\infty\)
  • The range of all logarithmic functions is: \(-\infty<x<\infty\)
  • The only transformation that affects the domain or range of the logarithmic function is a number added “inside” the function which causes a right (-) or left (+) translation of the function and its vertical asymptote. This shift marks the new left boundary of the domain with the right boundary remaining infinity. The range never changes.
  • If there is a number in front of the variable “inside” the function, all the terms inside must be divided by that number before deciding the new location of the vertical asymptote.

Video Highlights

  • 00:00 Introduction
  • 00:10 Domain and range of log parents
  • 03:20 Domain and range after a vertical translation
  • 05:07 Domain and range after a horizontal translation
  • 07:05 Domain and range with a horizontal stretch/compression
  • 08:35 Domain and range with a vertical stretch/compression
  • 11:37 Conclusion
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  • To watch this video on YouTube in a new window with clickable highlights, click here

Related Videos

Rewriting Logs to Exponential Form
Solving Log Equations by Rewriting in Exponential Form
Four Properties of Logs
Using Properties of Logs to Solve Log Equations
Solving Logs of All Bases with a Calculator
Solving Log Equations with a Calculator
Using the Inverse to Graph the Log Parent Function
Graphing More Complex Log Functions

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