Equation for exponential function with covariate I understand that the formula for an exponential relationship between X and Y can be written as follows:
Y = a*exp(b*X) = exp(a+b*X)
And from there, if I want to run as a standard regression, I would take the logarithm of Y and then regression ln(Y) on X:
ln(Y) = a + b*X
I was wondering what the extension of this would be if I wanted to include a covariate. Would it be:
Y = a*exp(b1*X1)+b2*X2
And, could I then run a regression of the form:
ln(Y) = a + b1*X1 + b2*X2
 A: Your initial equation should have been: $\exp(a) \times \exp(bX) = \exp(a+ bX)$, so the extension to multiple variables  would be 
$exp(a) \times \exp(b_1X_1) \times \exp(b_2X_2)$. 
However, in general I would recommend against log transforming the dependent variable, but instead use a model with a log link function. That way you have the advantage of the logarithm and still model the mean of the original dependent variable (the latter is no longer true if you take the logarithm of the dependent variable).  Here are some references to look at for this point:
Nicholas J. Cox, Jeff Warburton, Alona Armstrong, Victoria J. Holliday (2007) "Fitting concentration and load rating curves with generalized linear models" Earth Surface Processes and Landforms, 33(1):25--39. link
Santos Silva, J.M.C., S. Tenreyro 2006. "The log of gravity", The Review of Economics and Statistics 88(4):641-658. link
Jeffery Wooldridge, Econometric Analysis of Cross Section and Panel Data, 2nd ed. MIT Press, Chapter 18.
