How do I solve this velocity equation?

Question

$$t-t_0=m\int\frac{\mathrm dv}{F}=-\frac{m}{b}\int\frac{\mathrm dv}{v}$$ $$t-t_0=-\frac{m}{b}\ln\frac{v}{v_0}$$ $$v=v_0 \cdot e^{-\frac{b}{m}(t-t_0)}$$

Can you please help me? How do I convert it from $\ln$ into exponential?

Answer 1

If you know what logarithms ARE you should know that logarithms and exponentials are inverse functions. $\log_a(a^x)= a^{\log_a(x)}= x$. In particular, the "natural logarithm" is defined as the inverse to $e^x$. $e^{\ln(x)}= \ln(e^x)= x$.

Given $t- t_0= -\frac{m}{b}\ln\left(\frac{v}{v_0}\right)$ Dividing both sides by $-m/b$, $-\frac{b}{m}(t- t_0)= \ln\left(\frac{v}{v_0}\right)$

Now take the exponential of both sides. That eliminates the logarithm on the right: $e^{-\frac{b}{m}(t- t_0)}= \frac{v}{v_0}$

Finally, multiply both sides by $v_0$ to get

$v= v_0e^{-\frac{b}{m}(t- t_0)}$