# How to linearize $V=V_w+(V_0-V_w)e^{-kt}$

I have a physics homework and I was asked to transform $v=v_w+(v_0-v_w)e^{-kt}$ into a linear equation to be graphed. ($v_w$ is one variable that is constant and $k$ is constant.) $v$ is velocity and $t$ is time. Usually $v_0$ is the initial velocity so it's also constant I'm guessing. There is no indication as to what we need to graph on the ordinate or abscissa.

We need to transform the equation as well as say what their $m$ value and $b$ value would be. Help! I've tried playing with natural $\log$ but couldn't get anything that was linear.

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There are two interesting regimes, $t\approx 0$ and $t\to\infty$. At early times, you have $$v(t)=v_w+(v_0-v_w)e^{-kt}\approx v_w+(v_0-v_w)(1-kt)=v_0-(v_0-v_w)kt.$$ I assume you can figure out the slope and intercept of that line.
As $t\to\infty$, the velocity asymptotes to the constant $v(t)\sim v_w$.