Question

Can someone tell me how they got from the first expression to the second one?

https://flic.kr/p/r44hXT

Answer 1

multiply `-b` through out
then change `ln` to exponent form

Answer 2

You need to know that:\[e ^{\ln(a)} = a\]
and
\[(e ^{a})(e ^{b})=e ^{{a+b}}\]
\[(e ^{a})(e ^{-b})=e ^{(a-b)}\]
Now solving the expression:
\[-(1/b)(\ln \left| mg-bv \right|)= t/m -c\]
\[\ln \left| mg-bv \right|=-b(t/m+c)|\]
Aplying euler for both sides
\[e ^{\ln \left| mg-bv \right|}=e ^{-b(t/m+c)}\]
Aplying the property shown:\[\left| mg-bv \right|=e ^{(-b)(t/m+c)}\]\
Solving the right side\[\left| mg-bv \right|=e ^{[(-bt/m) -(cb)]}\]
Applying the second property on the right side:
\[\left| mg-bv \right|=(e ^{-bt/m})(e ^{-cb})\]
Which is the solution, hope it helped you!