Problem

$73-78=$ Domain Find the domain of the function. …

Problem 77 Hard Difficulty

$73-78=$ Domain Find the domain of the function.
$$h(x)=\ln x+\ln (2-x)$$

Answer

$D_{h}=(0,2)$

Related Courses

Algebra

Algebra and Trigonometry

Chapter 4

Exponential and Logarithmic Functions

Section 3

Logarithmic Functions

Discussion

You must be signed in to discuss.

Top Algebra Educators

Watch More Solved Questions in Chapter 4

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

Problem 29

Problem 30

Problem 31

Problem 32

Problem 33

Problem 34

Problem 35

Problem 36

Problem 37

Problem 38

Problem 39

Problem 40

Problem 41

Problem 42

Problem 43

Problem 44

Problem 45

Problem 46

Problem 47

Problem 48

Problem 49

Problem 50

Problem 51

Problem 52

Problem 53

Problem 54

Problem 55

Problem 56

Problem 57

Problem 58

Problem 59

Problem 60

Problem 61

Problem 62

Problem 63

Problem 64

Problem 65

Problem 66

Problem 67

Problem 68

Problem 69

Problem 70

Problem 71

Problem 72

Problem 73

Problem 74

Problem 75

Problem 76

Problem 77

Problem 78

Problem 79

Problem 80

Problem 81

Problem 82

Problem 83

Problem 84

Problem 85

Problem 86

Problem 87

Problem 88

Problem 89

Problem 90

Problem 91

Problem 92

Problem 93

Problem 94

Problem 95

Problem 96

Problem 97

Problem 98

Problem 99

Problem 100

Problem 101

Problem 102

Problem 103

Problem 104

Problem 105

Video Transcript

All right. So our question for today is to find the domain of the function h of X equals Ellen of X plus island of two minus X. So to begin answering this question, we need to know what the domain of the function means. What does that mean? So the domain of a function means this is the employer or argument values for which a function is defined. Okay, so now that we know that we can begin to answer the question, actually, we needed a little more additional information. So, um, there's a few properties of natural logarithms, namely, the natural logarithms cannot be defined at zero. So natural log of zero is not defined as we have stated right here. Um, and also, natural logs cannot be the farm for negative numbers, so you can't have the natural log on negative one. They get 203 so on and so forth. We cannot have that because, um, the deafness and of the natural log is love based E. And we can't have, um he can never be anything negative. So we can have the natural log a negative number. So now that we know we can be instant. Answer the question. Um, so we don't have to recall the product world for logarithms in order to answer this question and product rule for logarithms not just ah, natural logarithms. It's also for, um, any based logarithms. We have that the lug of a ah and being arbitrary, constant plus the logo be be being another are arbitrary. Constant is equal to the log er a times B. So the product Ellen of a maybe sort of street. So now we know that we have a set of X People's Ellen of X plus Ellen of two minus X. This can be re written using the product roll for logarithms as island as h of X equals. I went of X times two minus x, and we can define a whole new, uh function and call this function G FX, which is just the argument argument being all of this in red here, inbred Hold on here of this and black, Actually, there we go. Yeah, so that's our argument and student. And well, to find the argument as a whole new function, we have Ellen of eggs or just the argument. So ex times two minus Nix. That's our argument. So we want to know when this is equal to zero. Because again, as stated previously, Ellen of zero is not to find. So in order to do that, we set This is equal to zero, and this is equal to zero where x exes, You go to zero and where X is equal to this is where G of X is equal to zero. And remember, we can't have equal to zero, and I cannot be a negative number. So ex cannot be, uh, zero. That's cannot be so. So let's practice here and blue Thanks, Canaanites be equal to zero. Thanks cannot be equal soon, so and additionally, we cannot have a negative argument, So X cannot be anything greater than soon. Ex cannot be anything less than zero. Because if we have anything greater than so we have, uh, three say, if we had three, we have two minus three, which is negative one three times negative One would be Ellen of negative three. So we can't have that. So our domain is where don't moon is X mix between zero and so less than zero were greater than zero and listening so So this is our domain. And that is our answer to the question. And just to verify that would look at a graph of the function. So we have only a little X minus X squared, which is the experience for me. Ah, patient X hearing, and as we can see, um, did the man is nuts exceeding zero, And soon And actually it's never equal deserve So So yeah, less Soon, too. And greater than zero. All right.

We have video lessons for 89.11% of the questions in this textbook