# Why is the deivative 2ln(x) is the same as that of ln(x^2)

1. Aug 22, 2011

### mahmoud2011

In the textbooks some times they simplify the the logarithmic using logarithmic properties . But sometimes the domain of the simplified form is not the same as that of not simplified so how their derivative are equal. ???????

2. Aug 22, 2011

### disregardthat

The domains of the derivatives are not equal.

3. Aug 22, 2011

### mahmoud2011

I know but mustn't we simplify and take the derivative ,right ?

4. Aug 22, 2011

### disregardthat

We don't have to, but we can. However note that 2log(x) is not a simplification of log(x^2) for negative x.

5. Aug 22, 2011

### mahmoud2011

That is what I am talking about.

6. Aug 22, 2011

### SteveL27

They are the same. The derivative of 2*ln(x) = 2/x.

The derivative of ln(x^2) = (1/x^2) * 2x = 2/x.

7. Aug 22, 2011

### uart

Yep I can see that Mahmoud.

Yes you make a good point. I sometimes see them treated as being identical when as you point out they are not.

$$\frac{d}{dx} \left( 2 \log_e(x) \right) = \frac{2}{x} \,\,\,\,\,\,:\,\,\,\, x>0$$

whereas

$$\frac{d}{dx} \left(\log_e(x^2) \right) = \frac{2}{x} \,\,\,\,\,\,:\,\,\,\, x \neq 0$$

8. Aug 22, 2011

### SteveL27

Isn't the point just that the original functions 2log(x) and log(x^2) have different domains? This question really has nothing to do with derivatives. But it's a good point ... you can't arbitrarily apply the log laws without double-checking the domains.

9. Aug 22, 2011

### mahmoud2011

Thanks for you all . Every time I use logarithmic laws I check the domains but my textbook doesn't mention that.

10. Aug 22, 2011

### Staff: Mentor

If your textbook simplifies ln(x2) to 2 ln(x), they are probably making the assumption that x > 0. If they don't show that assumption anywhere, then they are being very sloppy.

11. Aug 22, 2011

### mahmoud2011

No , this problem doesn't exist but I wanted to say that why the textbook didn't mention that I must check the domains , Is the author wants me to do it myself and actually I did.