1. ## logarithmic simplification.

Express the given quantity as a single logarithm.

and now for my attempt

oh, BTW, sqrt = square root. I dont know how to make the symbol on this forum.

$\displaystyle ln(1+x^6)-ln(cos(x))+(ln(x)/2)$

$\displaystyle ln((1+x^6)/(cos(x))+ln(x)/2$

$\displaystyle ln((1+x^6)/(cos(x))+ln(sqrt(x))$

$\displaystyle ln((1+x^6)/(cos(x)(ln(sqrt(x))$

$\displaystyle ln((sqrt(x)(x^6+1))/cos(x))$

man, I wish I could make this look nicer so its easier to read. But anyway, that last entry is what I got. I just want to be sure.

2. Hello,

Yup, that's correct !

As for the codes :

\frac{numerator}{denominator} gives $\displaystyle \frac{numerator}{denominator}$

\ln gives $\displaystyle \ln$ instead of $\displaystyle ln$. Same for \cos

\sqrt{bla} gives $\displaystyle \sqrt{bla}$

Sidenote : that's very good, you put the parenthesis where they needed to be !

3. you did fine ...

$\displaystyle \ln\left(\frac{\sqrt{x}(1+x^6)}{\cos{x}}\right)$

4. I second guess myself too much I guess.

5. Hello, leftyguitarjoe!

What you did is correct!
And thank you for showing your work!

Express as a single logarithm: .$\displaystyle \ln(1+x^6) + \frac{1}{2}\ln(x) - \ln(\cos(x))$
I would do it like this . . .

$\displaystyle \ln(1 + x^6) + \ln\left(x^{\frac{1}{2}}\right) - \ln(\cos(x))$

. . $\displaystyle = \;\ln\bigg[(1+x^6)x^{\frac{1}{2}}\bigg] - \ln(\cos(x))$

. . $\displaystyle = \;\ln\left[\frac{\sqrt{x}\,(1+x^6)}{\cos(x)}\right]$

6. I think its useless if I dont show my work. If I didnt, you wouldnt be able to see where I messed up if I had got it wrong.

7. Originally Posted by leftyguitarjoe
I think its useless if I dont show my work. If I didnt, you wouldnt be able to see where I messed up if I had got it wrong.
you wouldn't believe how many students here find that concept difficult to grasp! you deserve that thanks more than you know!