# Simplifying a log expression with identities

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1. Apr 20, 2016

### ProfuselyQuarky

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?

2. Apr 20, 2016

### Math_QED

Looks correct though. Note that Ln0 is undefined, so you might want to specify when the equality is correct.

3. Apr 20, 2016

### ProfuselyQuarky

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. Apr 20, 2016

### ProfuselyQuarky

So what is the error?

5. Apr 20, 2016

### SteamKing

Staff Emeritus
You sure about that? You mean cos x is never equal to zero?

6. Apr 20, 2016

### ProfuselyQuarky

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. Apr 20, 2016

### ProfuselyQuarky

But would that affect the answer? How did I simplify wrong?

8. Apr 20, 2016

### SteamKing

Staff Emeritus
So what happens to ln |cos x| when x = (2k+1)π/2 ?

9. Apr 20, 2016

### ProfuselyQuarky

We get ln 0 which is undefined

10. Apr 20, 2016

### ProfuselyQuarky

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. Apr 20, 2016

### Math_QED

The answer is correct, as long as you specify for which values the expression is undefined.

12. Apr 20, 2016

### SammyS

Staff Emeritus
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.

13. Apr 20, 2016

### Math_QED

Why do you think your expression is wrong in the first place?

14. Apr 20, 2016

### ProfuselyQuarky

It was marked wrong.

15. Apr 20, 2016

### SammyS

Staff Emeritus
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. Apr 20, 2016

### Math_QED

Even then, it shouldn't be marked wrong since the expression itself is correct.

17. Apr 20, 2016

### ehild

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.