Ex: Find the Domain of Logarithmic Functions — Transcript

Full transcript

1. 0:00We want to find the domain of the given log  functions and then state the domain using
2. 0:04inequalities and interval notation. Looking at  our first function, we have y equals log base two
3. 0:11of the quantity two x minus four. The quantity  two x minus four is the number part of the log
4. 0:17that's obtained by raising two to the power of y.  Remember, in exponential form, two is the base,
5. 0:25y is the exponent, and the number  would be the quantity two x minus four.
6. 0:35So if we think about all the positive  values we get when raising two to a power,
7. 0:40well, the result is always  going to be greater than zero.
8. 0:43That's the reason why to find the domain,  we have to solve the inequality two x minus
9. 0:49four is greater than zero. The number part of a  logarithm is always going to be greater than zero.
10. 0:56So now we'll add four to both sides,  which would give us two x is greater
11. 1:00than four. Now we divide both sides by two, so we  have x is greater than two. Therefore, our domain
12. 1:11would be x greater than two.
13. 1:13Using inequalities, if we wanted to express this  using interval notation, it might be helpful to
14. 1:19sketch a graph. So if this is two, we'd have  an open point on two and an arrow to the right
15. 1:26approaching positive infinity. Therefore, using  interval notation, we have the interval from
16. 1:31two to infinity, and it's open on  two because it does not include two.
17. 1:38Okay, looking at our second example, notice  how there's no base listed on this log.
18. 1:43Therefore, we know it's common log or log base
19. 1:48ten. Again, if we focus on the number part of the  log, the quantity x squared minus three x would be
20. 1:54obtained by raising ten to the power of y, which  would always be greater than zero. Therefore,
21. 2:00to find the domain of this function, we need  to solve the inequality x squared minus three x
22. 2:06greater than zero. The solutions to  this will be the domain of our function.
23. 2:12To solve a quadratic inequality,  we first find the solutions
24. 2:16as if it was an equation. This does factor,  so we want to first solve the equation
25. 2:20x times the quantity x minus three is equal  to zero. This would be zero when x equals zero
26. 2:28or when x equals three. So these values of  x squared would be equal to zero. We want to
27. 2:35find the x values where it's greater than zero.  So what we'll do now is sketch a number line.
28. 2:42Plot these two values as open points  because of the inequality symbol.
29. 2:49Then we will test x values in each  of these three intervals to see
30. 2:53which satisfy the original inequality.  So let's test x equals negative one,
31. 3:00let's say one, and let's say four. If we  let x equal four, we'd have four squared
32. 3:07is sixteen minus twelve, which is four, and that  is greater than zero, so this interval is true.
33. 3:17Let's go ahead and graph it  approaching positive infinity.
34. 3:23When x is one, we'd have one minus three,  which is negative two, and that's not
35. 3:28greater than zero, so this is false. When  x is negative one, we'd have positive one
36. 3:36minus three times negative one, which becomes  one plus three, which is four, and that's greater
37. 3:42than zero, so this is true. This interval is also  part of the domain approaching negative infinity.
38. 3:52So the domain using inequalities would be x is  less than zero or x is greater than three. When
39. 4:05we are using interval notation, we would have  the interval from negative infinity to zero
40. 4:12and the interval from three to infinity. Again,
41. 4:16it does not include these endpoints, therefore  we have open intervals on three and zero.
42. 4:27Okay, thank you for watching.