simplifying 1+lnx(1) - x *1/x all divided by (1+ln x)2

[kmce]

So i think this is algebra. We arent taught it as algebra, calculus etc, its all just math, so i hope i am in the right place.

So i have worked out a calculation to equal 1+lnx(1) - x *1/x all divided by (1+ln x)²

but i need to simply it to ln x / (ln x +1)²

Can anyone help with how to do this

 

[MarkFL]

You want to show:

\(\displaystyle \displaystyle \frac{1+\ln(x)(1)-x\cdot\dfrac{1}{x}}{(1+\ln(x))^2}= \frac{\ln(x)}{(\ln(x)+1)^2}\)

According to the Multiplicative Identity Property, \(\displaystyle 1\cdot a=a\), so the numerator on the LHS becomes:

\(\displaystyle \displaystyle \frac{1+\ln(x)-x\cdot\dfrac{1}{x}}{(1+\ln(x))^2}= \frac{\ln(x)}{(\ln(x)+1)^2}\)

To continue:

\(\displaystyle \displaystyle x\cdot\frac{1}{x}=\frac{x}{x}=\,?\)

 

[kmce]

Ok, i think i get it. so the x * 1/x would technically become 1x/x so would just cancel out?

Not entirely sure where the 1+ goes to though.

Thanks for all the help btw.

 

[MarkFL]

Ok, i think i get it. so the x * 1/x would technically become 1x/x so would just cancel out?

Not entirely sure where the 1+ goes to though.

Thanks for all the help btw.

Yes, \(\displaystyle \displaystyle \frac{x}{x}=1\)

And so we now have:

\(\displaystyle \displaystyle \frac{1+\ln(x)-1}{(1+\ln(x))^2}= \frac{\ln(x)}{(\ln(x)+1)^2}\)

Now, we may use the commutative and associative properties of addition, i.e.:

\(\displaystyle a+b=b+a\)

\(\displaystyle a+b+c=(a+b)+c\)

And we may rewrite the numerator on the LHS as follows:

\(\displaystyle \displaystyle \frac{(1-1)+\ln(x)}{(1+\ln(x))^2}= \frac{\ln(x)}{(\ln(x)+1)^2}\)

Can you continue?

 

[kmce]

oh ok, so the 1x/x = 1 so the 1-1 would cancel each other out, given ln(x) and then just the bottom is just rearranged for some reason.

I has been over a decade since i have done math so i apologise if my question are really stupid

 

[MarkFL]

oh ok, so the 1x/x = 1 so the 1-1 would cancel each other out, given ln(x) and then just the bottom is just rearranged for some reason.

I has been over a decade since i have done math so i apologise if my question are really stupid

The only "not so smart" thing is to not try to gain a better understanding.