Simplifying a log expression with identities

In summary: Even then, it shouldn't be marked wrong since the expression itself...In summary, the expression ##\ln |\cot {x}|+\ln |\tan {x}\cos {x}|## is undefined for some values of x, but it is defined for ln(cos(x)) .
  • #1

ProfuselyQuarky

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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?
 
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  • #2
ProfuselyQuarky said:
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
Math_QED said:
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
So what is the error?
 
  • #5
ProfuselyQuarky said:
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?
 
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  • #6
SteamKing said:
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
But would that affect the answer? How did I simplify wrong?
 
  • #8
ProfuselyQuarky said:
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
SteamKing said:
So what happens to ln |cos x| when x = (2k+1)π/2 ?
We get ln 0 which is undefined
 
  • #10
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
ProfuselyQuarky said:
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.
 
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  • #12
ProfuselyQuarky said:
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.
 
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  • #13
Why do you think your expression is wrong in the first place?
 
  • #14
Math_QED said:
Why do you think your expression is wrong in the first place?
It was marked wrong.
 
  • #15
ProfuselyQuarky said:
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
SammyS said:
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
Math_QED said:
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.
 

1. How do I simplify a log expression using identities?

To simplify a log expression using identities, you can start by using the basic logarithm identities, such as log(ab) = log(a) + log(b) and log(a/b) = log(a) - log(b). Then, you can use the power rule, product rule, and quotient rule to further simplify the expression.

2. Can I use identities to simplify any log expression?

Yes, identities can be used to simplify any log expression. However, it may not always result in a simplified expression, as some log expressions are already in their simplest form.

3. What is the purpose of simplifying a log expression with identities?

The purpose of simplifying a log expression with identities is to make the expression easier to work with and understand. It can also help to identify patterns and relationships between different logarithmic expressions.

4. Are there any specific steps to follow when simplifying a log expression with identities?

Yes, there are some specific steps to follow when simplifying a log expression with identities. These include using the basic logarithm identities, applying the power rule, product rule, and quotient rule, and combining like terms to simplify the expression as much as possible.

5. Can I use identities to simplify a log expression with variables?

Yes, identities can be used to simplify a log expression with variables. In fact, using identities can be particularly helpful when dealing with expressions that involve variables, as it can help to identify patterns and simplify the expression algebraically.

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