How do you solve this expression for T? 2=e^(.003t)
The answer is t=ln2/.003
In general, to "solve" an equation for t, you do, to both sides, the "opposite" of what is done to t, in the reverse order. Here, $\displaystyle y= e^{.003t}$. To evaluate that, you would do two things: first multiply t by 0.003, then take the exponential. We need to do the "opposite" (inverse) of the exponential, then the "opposite" of "multiply by 0.003". The inverse of the exponential function is the natural logarithm and the inverse of "multiply by 0.003" is "divide by 0.003".
So we would start with
$\displaystyle y= e^{.003t}$ and first take the logarithm of both sides: $\displaystyle ln(y)= .003t$.
Now, divide both sides by .003: $\displaystyle \frac{ln(y)}{.003}= t$.