# Relation between log and arcs

1. Nov 14, 2013

### Jhenrique

If there is a formula relating the exponential with sine and cosine normal and hyperbolic (exp(ix) = cos(x) + i sin(x), exp(x) = cosh(x) + sinh(x)), there is also a formula relating the logarithm with arcsin, arccos, and arcsinh arccosh?

2. Nov 14, 2013

### SteamKing

Staff Emeritus
3. Nov 14, 2013

### Jhenrique

But, But I wonder if there is a general expression that combines the sine the cosine (hyperbolic or no) of one side of the equality with the logarithm, in other side of the equality...

4. Nov 14, 2013

### SteamKing

Staff Emeritus
Are you talking about expressing sin(x) and cos(x) in terms of log(x)? Your question is not very clear.

5. Nov 14, 2013

### Jhenrique

Sorry. I'm talking about an expression of log(x) in terms of arcsinh(x) and arcosh(x).

6. Nov 14, 2013

### SteamKing

Staff Emeritus

7. Nov 14, 2013

### Jhenrique

Last edited by a moderator: May 6, 2017
8. Nov 14, 2013

### SteamKing

Staff Emeritus
You mean ln (e^x) = x and ln (e^ix) = ix? I'm sorry, I'm not following your question.

9. Nov 14, 2013

### Jhenrique

I apologize too, because my English is primitive...

I'm trying to say that if you can combine sine and cosine to express the exponential, then it should also be possible to combine and arcsin arccos to express the logarithm. But this combination is not so simple ... I tried to add and multiply arcsineh(x) with arccosh(x), I tried to combine they by arithmetic and geometric mean, I tried to break log(x) on even and odd function. I've tried several things, but I was not able to find a true expression.

I look for an expression as log(x) = arccosh(x) + arcsinh(x). This expression is false, but it is close of the genuine.

10. Nov 14, 2013

### lurflurf

Do you mean

$$\log(x)=\mathrm{arcsinh} \left( \frac{x^2-1}{2x} \right)=\imath \arcsin \left( \frac{x^2-1}{2 \imath x} \right)$$

This only holds for 0<x
but similar expressions can be used for x complex