Writing Natural Log in an e Form

[Jason76]

Here is the general solution to a Differential Equations problem:

\(\displaystyle \ln(y + 1) = \ln(x) + C\)

but this can be made more simple using e.

For instance the right hand side becomes \(\displaystyle e^{\ln(x) + C}\)

What does the left side become and why?

 

[HallsofIvy]

Here is the general solution to a Differential Equations problem:

\(\displaystyle \ln(y + 1) = \ln(x) + C\)

but this can be made more simple using e.

For instance the right hand side becomes \(\displaystyle e^{\ln(x) + C}\)

What does the left side become and why?

You are taking the exponential of both sides: that is why ln(x)+ C becomes \(\displaystyle e^{ln(x)+ C}\). In fact, that can be further simplified:
\(\displaystyle e^{ln(x)+ C}= e^{ln(x)}e^{C}= cx\) because \(\displaystyle e^{ln(x)}= x\) (e^x and ln(x) are inverse functions and I have defined c= e^C).

On the left, taking the exponential f bth sides, \(\displaystyle e^{ln(y+1})= y+ 1\), again because \(\displaystyle e^{ln(x)}= x\)

So \(\displaystyle ln(y+ 1)= ln(x)+ C\) becomes y+ 1= cx or y= cx- 1.