Simple question regarding 1st order linear ODEs

Question

In a proof of the general solution to $y' = ay$ with $a \in \mathbb{R}$ the following implication is used.

$$\frac{y'}{y} = a \implies \ln(| y |)' = a$$

How is this derived?

Answer 1

Consider the derivative of $\ln \vert f(x)\vert $, which is $\frac{f'(x)}{f(x)}$. If you replace $f(x)$ with $y$, can you see how the result is derived

Answer 2

For $y>0$

$$(\ln |y|)’=(\ln y)’=\frac {y’}{y}$$

for $y<0$

$$(\ln |y|)’=(\ln -y)’=\frac {-y’}{-y}=\frac {y’}{y}$$

therefore

$$(\ln |y|)’=\frac {y’}{y}$$

Answer 3

Apply the chain rule $$(\ln y)'=\frac {d \ln y}{dx}=\frac {d \ln y}{dy}\frac {dy}{dx}=\frac 1 y\frac {dy}{dx}=\frac {y'}{y}$$