# Thread: Log Qn (Not too hard!!)

1. ## Log Qn (Not too hard!!)

Show that

2. Originally Posted by matty888
Show that

using the integral defintion of the natural log we get

$\ln(x)=\int_{1}^{x}\frac{1}{t}dt$

let $u=t^{1/n} \iff u^n=t \to nu^{n-1}=dt$

$\ln(x^n)=\int_{1}^{x^n}\frac{1}{t}dt=\int_{1}^{x}\ frac{1}{u^n}(nu^{n-1}du)=n\int_{1}^{x}\frac{1}{u}du=n\ln(x)$

3. ## Thank you(Further Qn)

TheEmptySet you are a lifesaver,I have one further question,could you tell me how the limits of integration change from 1 and x^n to 1 and u? I would be very grateful(I kinda know its to do with the fact that you made a substitution)!

4. Originally Posted by matty888
TheEmptySet you are a lifesaver,I have one further question,could you tell me how the limits of integration change from 1 and x^n to 1 and u? I would be very grateful(I kinda know its to do with the fact that you made a substitution)!
$u=t^{\frac{1}{n}}$ or u as a function of t

$u(x^n)=t^{\frac{1}{n}} \to u(x^n)=(x^n)^{\frac{1}{n}}=x$

it is a typo. It should be x.

I will fix the above post.