Simplifying a log expression with identities

  • #1
ProfuselyQuarky
Gold Member
817
527
I was supposed to simplify the expression ##\ln |\cot {x}|+\ln |\tan {x}\cos {x}|## and apparently it’s wrong. Where’s the mistake? Is it not simplified enough or . . . ?

##\ln |\cot {x}|+\ln |\tan {x}\cos {x}|##

##=\ln |\frac {\cos {x}}{\sin {x}}|+\ln |\frac {\sin {x}}{\cos {x}}\cdot \cos {x}|##

##=\ln |\frac {\cos {x}}{\sin {x}}|+\ln |\sin {x}|##

##=\ln |\frac {\cos {x}\sin {x}}{\sin {x}}|##

##=\ln |\cos {x}|##

If it’s really obvious, I’ll be happy with a hint. I thought that this was easy, but apparently I was mistaken?
 

Answers and Replies

  • #2
member 587159
I was supposed to simplify the expression ##\ln |\cot {x}|+\ln |\tan {x}\cos {x}|## and apparently it’s wrong. Where’s the mistake? Is it not simplified enough or . . . ?

##\ln |\cot {x}|+\ln |\tan {x}\cos {x}|##

##=\ln |\frac {\cos {x}}{\sin {x}}|+\ln |\frac {\sin {x}}{\cos {x}}\cdot \cos {x}|##

##=\ln |\frac {\cos {x}}{\sin {x}}|+\ln |\sin {x}|##

##=\ln |\frac {\cos {x}\sin {x}}{\sin {x}}|##

##=\ln |\cos {x}|##

If it’s really obvious, I’ll be happy with a hint. I thought that this was easy, but apparently I was mistaken?
Looks correct though. Note that Ln0 is undefined, so you might want to specify when the equality is correct.
 
  • #3
ProfuselyQuarky
Gold Member
817
527
Looks correct though. Note that Ln0 is undefined, so you might want to specify when the equality is correct.
Well, I was given the problem with no specifications and would it really affect the answer? I mean, this expression doesn’t deal with ##\ln {0}##.
 
  • #4
ProfuselyQuarky
Gold Member
817
527
So what is the error?
 
  • #5
SteamKing
Staff Emeritus
Science Advisor
Homework Helper
12,796
1,668
Well, I was given the problem with no specifications and would it really affect the answer? I mean, this expression doesn’t deal with ##\ln {0}##.
You sure about that? You mean cos x is never equal to zero?
 
  • Like
Likes ProfuselyQuarky
  • #6
ProfuselyQuarky
Gold Member
817
527
You sure about that? You mean cos x is never equal to zero?
Oh, I never said cos x never is equal to zero. ##\cos {\frac {\pi}{2}}=0## and ##\cos {\frac {3\pi}{2}}=0## and so on.
 
  • #7
ProfuselyQuarky
Gold Member
817
527
But would that affect the answer? How did I simplify wrong?
 
  • #8
SteamKing
Staff Emeritus
Science Advisor
Homework Helper
12,796
1,668
Oh, I never said cos x never is equal to zero. ##\cos {\frac {\pi}{2}}=0## and ##\cos {\frac {3\pi}{2}}=0## and so on.
So what happens to ln |cos x| when x = (2k+1)π/2 ?
 
  • #9
ProfuselyQuarky
Gold Member
817
527
So what happens to ln |cos x| when x = (2k+1)π/2 ?
We get ln 0 which is undefined
 
  • #10
ProfuselyQuarky
Gold Member
817
527
Oh, wait, so I'd have to specify that ##\ln |\cos {x}|## is only the answer when ##\cos {x}\neq0##.

That's great but there wasn't a single example of any of that in my book which is why I was thinking that my error had something to do with using the identities wrong.
 
  • #11
member 587159
Oh, wait, so I'd have to specify that ##\ln |\cos {x}|## is only the answer when ##\cos {x}\neq0##.

That's great but there wasn't a single example of any of that in my book which is why I was thinking that my error had something to do with using the identities wrong.
The answer is correct, as long as you specify for which values the expression is undefined.
 
  • Like
Likes ProfuselyQuarky
  • #12
SammyS
Staff Emeritus
Science Advisor
Homework Helper
Gold Member
11,359
1,032
Oh, I never said cos x never is equal to zero. ##\cos {\frac {\pi}{2}}=0## and ##\cos {\frac {3\pi}{2}}=0## and so on.
Frankly. I don't think this is the problem. After all, the original expression isn't defined for values of x which make cos(x)=0, either.

However, there are values of x for which the original expression is undefined, but are defined for ln(cos(x)) . Throw those out.
 
  • Like
Likes ProfuselyQuarky
  • #13
member 587159
Why do you think your expression is wrong in the first place?
 
  • #14
ProfuselyQuarky
Gold Member
817
527
Why do you think your expression is wrong in the first place?
It was marked wrong.
 
  • #15
SammyS
Staff Emeritus
Science Advisor
Homework Helper
Gold Member
11,359
1,032
Oh, wait, so I'd have to specify that ##\ln |\cos {x}|## is only the answer when ##\cos {x}\neq0##.

That's great but there wasn't a single example of any of that in my book which is why I was thinking that my error had something to do with using the identities wrong.
I'm not sure I was very clear in my previous post.

I don't think the problem is:
that ##\ln |\cos {x}|## is only the answer when ##\cos {x}\neq0##.​
Those values of x at which ##\cos {x}=0## are not in the domain of ## \ \ln |\cot {x}|+\ln |\tan {x}\cos {x}|\ ## in the first place.

I think that the problem is that ##\ln |\cos {x}|## is defined for some values of x for which ## \ \ln |\cot {x}|+\ln |\tan {x}\cos {x}|\ ## is not defined. I think that you must restrict the domain of the answer to eliminate those.
 
  • #16
member 587159
I'm not sure I was very clear in my previous post.

I don't think the problem is:
that ##\ln |\cos {x}|## is only the answer when ##\cos {x}\neq0##.​
Those values of x at which ##\cos {x}=0## are not in the domain of ## \ \ln |\cot {x}|+\ln |\tan {x}\cos {x}|\ ## in the first place.

I think that the problem is that ##\ln |\cos {x}|## is defined for some values of x for which ## \ \ln |\cot {x}|+\ln |\tan {x}\cos {x}|\ ## is not defined. I think that you must restrict the domain of the answer to eliminate those.
Even then, it shouldn't be marked wrong since the expression itself is correct.
 
  • #17
ehild
Homework Helper
15,543
1,909
Even then, it shouldn't be marked wrong since the expression itself is correct.
SammyS was right. The initial and final expressions are not identical, as their ranges are different. The final one is not defined when cos(x)=0. The initial one is not defined when either sin(x) or cos(x) is zero.
 

Related Threads on Simplifying a log expression with identities

  • Last Post
Replies
16
Views
4K
Replies
27
Views
9K
Replies
7
Views
8K
Replies
14
Views
6K
  • Last Post
Replies
3
Views
755
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
2
Views
1K
Top