Question regarding changing from natural log to e

Question

$$-\ln\left| 150 - y\right|=\frac{t}{200}-\ln130$$ $$\left| 150 - y\right|=130e^{-t/200}$$

I got $-130e^{t/200}$

Why is $e$ positive and the fraction negative?

Answer 1

From $$ -\ln\left| 150 - y\right|=\frac{t}{200}-\ln130 $$ you have, by multiplying by $-1$, $$ \ln\left| 150 - y\right|=\ln130-\frac{t}{200} $$ giving by exponentiation $$ \left| 150 - y\right|=e^{\large \ln130-\frac{t}{200}}=e^{\ln130}\cdot e^{\large -\frac{t}{200}}=130\cdot e^{\large -\frac{t}{200}} $$ as given.

Answer 2

it is $$\ln\left(\frac{130}{|150-y|}\right)=e^{t/200}$$ and then by multiplication with $$e^{-t/200}$$ we get $$130\cdot e^{-t/200}=|150-y|$$