simplify natural logarithmHow to simplify this natural logarithm12ln|y+1|−12ln|y−1|+ln|C|=12ln|x+1|−12ln|x−1|if I apply the logarithm ruleln⁡|y+1|−ln⁡|y−1|+12ln⁡(C2)=ln⁡|x+1|−ln⁡|x−1|Please help further..

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Eliza Flores · Accepted Answer

First, Note that x,y≠±1 and ln⁡C=12ln⁡C2. Denote C2=k&amp;gt;0. So the equation becomes 12(ln|y+1|−ln|y−1|+ln⁡k)=12(ln|x+1|−ln|x−1|)Cancelling 12 and using basic logarithm identities, we get,ln⁡(k|y+1y−1|)=ln⁡(|x+1x−1|)Now, we get,k|y+1y−1|=|x+1x−1|Now simplify it. Best way is to break into different cases.Case 1: |y|&amp;gt;1 and |x|&amp;gt;1Givesky+1y−1=x+1x−1Solve 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&#039;ll write the facts I used;ln⁡a+ln⁡b=ln⁡abln⁡a−ln⁡b=ln⁡abln⁡a=ln⁡b⇔a=ball of these holds for a,b∈R+

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

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