1. ## Natural Logarithms

In my answer book, I noticed that one of the steps were this:

How did the first step lead to the second step? Can someone help me explain how?

2. $\displaystyle x\ln(2x+1) + \frac{1}{2}\ln(2x+1) =$

factor out $\displaystyle \ln(2x+1)$ ...

$\displaystyle \ln(2x+1) \cdot \left(x + \frac{1}{2}\right) =$

change the appearance of $\displaystyle \left(x + \frac{1}{2}\right)$

$\displaystyle \ln(2x+1) \cdot \frac{1}{2}(2x+1)$

3. change the appearance of $\displaystyle \left(x + \frac{1}{2}\right)$

$\displaystyle \ln(2x+1) \cdot \frac{1}{2}(2x+1)$
Thanks for responding, but what do you mean by "change the appearance"?

4. $\displaystyle \left(x+\frac{1}{2}\right) = \frac{1}{2}(2x+1)$

5. Originally Posted by skeeter
$\displaystyle \left(x+\frac{1}{2}\right) = \frac{1}{2}(2x+1)$
But how did that first part turn into the second part?
EDIT: Nevermind, I got it now, thank you!

6. He just used logic to rewrite it in terms of the other polynomial.

Distribute that out.

$\displaystyle \frac{1}{2} * 2x = x$

and

$\displaystyle \frac{1}{2} * 1 = \frac{1}{2}$

so

$\displaystyle \frac{1}{2}(2x + 1) = (x + \frac{1}{2})$