simplify natural logarithm How to simplify this natural logarithm \frac{1}{2}\ln|y+1|-\frac{1}{2}\ln|y-1|+\ln|C| =\frac{1}{2}\ln|x+1|-\frac{1}{2}\ln|x-1| if I

caldaridvq

caldaridvq

Answered question

2022-04-26

simplify natural logarithm
How to simplify this natural logarithm
12ln|y+1|12ln|y1|+ln|C|=12ln|x+1|12ln|x1|
if I apply the logarithm rule
ln|y+1|ln|y1|+12ln(C2)=ln|x+1|ln|x1|
Please help further..

Answer & Explanation

Eliza Flores

Eliza Flores

Beginner2022-04-27Added 16 answers

First, Note that x,y±1 and lnC=12lnC2. Denote C2=k>0. So the equation becomes
12(ln|y+1|ln|y1|+lnk)=12(ln|x+1|ln|x1|)
Cancelling 12 and using basic logarithm identities, we get,
ln(k|y+1y1|)=ln(|x+1x1|)
Now, we get,
k|y+1y1|=|x+1x1|
Now simplify it. Best way is to break into different cases.
Case 1: |y|>1 and |x|>1
Gives
ky+1y1=x+1x1
Solve for y in term of x or vice versa as desired.
Other cases can be dealt with similarily.
Note: Seems like you are starting to learn about logarithms, so I'll write the facts I used;
lna+lnb=lnab
lnalnb=lnab
lna=lnba=b
all of these holds for a,bR+

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