aid with rewriting absolute values and converting log form to exponents

my first problem is to rewrite |x-2|+|3-x| given that 0<x<2

What I've done so far is that given the values possible for x, the x-2 portion is negative making it:

(-x+2)

with the same reasoning the 3-x portion is always positive keeping it as is so I come up with:

(-x+2)+(3-x)

and once I simplify I get 5-2x, is this correct or did I do a step wrong?

the second problem I have is rewriting ln(sqrt(e)) = 1/2 to exponential form

What I did here was rewrite the sqrt(e) to e^1/2, then bring down the 1/2 via the ln(a^b) = b ln (a) rule and I'm stuck at

1/2 ln(e) = 1/2

I'm not sure what I do now to get it to exponential form, the ln and sqrt e get me confused, any help on what I'm supposed to do?

Thanks

Re: aid with rewriting absolute values and converting log form to exponents

Yes the first one is correct.

As for the second, you seem to be misunderstanding what the question is asking of you. You are not asked to SOLVE an equation, you are asked to SHOW that the two sides of the equation are in fact the same thing.

$\displaystyle \displaystyle \begin{align*} \ln{ \left( \sqrt{e} \right) } &= \ln {\left( e^{\frac{1}{2}} \right) } \\ &= \frac{1}{2}\ln{(e)} \\ &= \frac{1}{2} \cdot 1 \\ &= \frac{1}{2} \end{align*}$

Re: aid with rewriting absolute values and converting log form to exponents

For the first problem, you are absolutely right. The second problem would just be to use the definition of logarithms to change the natural log back to exponents. All ln is is log with a base of e. If you don't know what e is, just know that it is a special mathematical constant you will encounter a lot in the future.