Natural Logs or Ln "lawn", collecting like terms.

Hi, this is my first post asking for a little knowledge concerning Natural Logs or, Ln "lawn". I Saved everyone a headache of me attempting to type out the question and answers as they could appear in text format on the forum, so i wrote up my question on paper and created a down scrolling album where u can easily see my question and hopefully be able to help me more easily seeing the actual question for what it is.

Natural Logs - Imgur

If you take that link you will find an album where it just shows pictures of my question in sequence, I am basically asking if there is anyone who can explain to me how to collect the like terms in this math question. You may have to right click the link and open in a new tab/window.

Thanks, Comedia

Re: Natural Logs or Ln "lawn", collecting like terms.

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Originally Posted by

**Comedia** Hi, this is my first post asking for a little knowledge concerning Natural Logs or, Ln "lawn".

If you take that link you will find an album where it just shows pictures of my question in sequence, I am basically asking if there is anyone who can explain to me how to collect the like terms in this math question.

First of all a "lawn" is area of grass with other plantings.

The trend in calculus textbooks is to use $\displaystyle \log(x)$ where we once would have used $\displaystyle \ln(x).$

Secondly, there is a web recourse.

Click it and you will see it helps with partial fractions beautifully.

Now once you have the parts, I would make three different integrals.

In general $\displaystyle \int_a^b {\frac{{f'(x)}}{{f(x)}}} dx = \log (b) - \log (a) = \log \left( {\frac{b}{a}} \right)$, of course $\displaystyle a~\&~b>0$.

So your first one is $\displaystyle \int_1^2 {\frac{{2dy}}{y}} = \left. {2\log (y)} \right|_1^2 = \log (4)$

Re: Natural Logs or Ln "lawn", collecting like terms.

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Originally Posted by

**Plato** First of all a "lawn" is area of grass with other plantings.[/TEX]

Yeah i just put that in for convenience just to freshen someones memory, i didnt think it was offensive haha.

My school seems to be behind on which textbook is preferred so all the examples are with ln(x) instead of log(x), and also its what I've learned, but ill try to conform to this, it seems easier get away from ln(x) anyways so its okay with me.

Also thanks for the input i think itll be easier for me to solve like this.

I had one other question about whether or not the anti derivative becomes the absolute value of Log|x| or just log(x) because it seems to flip between the two and for some reason i dont know when its supposed to be one or the other.

Re: Natural Logs or Ln "lawn", collecting like terms.

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Originally Posted by

**Comedia** I had one other question about whether or not the anti derivative becomes the absolute value of Log|x| or just log(x) because it seems to flip between the two and for some reason i dont know when its supposed to be one or the other.

That is correct. In fact you need to use $\displaystyle \ln(|y-3|)$ on the third integral because $\displaystyle (y-3)<0$ on $\displaystyle (1,2).$

Re: Natural Logs or Ln "lawn", collecting like terms.

Quote:

Originally Posted by

**Plato** First of all a "lawn" is area of grass with other plantings.

The trend in calculus textbooks is to use $\displaystyle \log(x)$ where we once would have used $\displaystyle \ln(x).$

Secondly, there is a

web recourse.

Click it and you will see it helps with partial fractions beautifully.

Now once you have the parts, I would make three different integrals.

In general $\displaystyle \int_a^b {\frac{{f'(x)}}{{f(x)}}} dx = \log (b) - \log (a) = \log \left( {\frac{b}{a}} \right)$, of course $\displaystyle a~\&~b>0$.

This is wrong. $\displaystyle \int_a^b {\frac{f'(x)}{f(x))}} dx= \log(f(b)- \log(f(a))= \log \left(\frac{f(b)}{f(a)}}\right)$

Quote:

So your first one is $\displaystyle \int_1^2 {\frac{{2dy}}{y}} = \left. {2\log (y)} \right|_1^2 = \log (4)$

Re: Natural Logs or Ln "lawn", collecting like terms.

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Originally Posted by

**HallsofIvy** This is wrong. $\displaystyle \int_a^b {\frac{f'(x)}{f(x))}} dx= \log(f(b)- \log(f(a))= \log \left(\frac{f(b)}{f(a)}}\right)$

Only because its missing a ) and should use absolute value.

$\displaystyle \int_a^b {\frac{f'(x)}{f(x)}} dx= \log(|f(b)|)- \log(|f(a)|)= \log \left(\frac{|f(b)|}{|f(a)|}}\right)$