Beta as an exponent? Can beta be an exponent within a function prior to taking the log of that function to arrive at a simple, log-linear estimation equation?

I am curious because I found an equation that had beta as an exponent on page 7 of this article by Lee Branstetter(2004): http://bit.ly/1OGSFx9 titled: Is Foreign Direct Investment a Channel of Knowledge Spillovers? Evidence from Japan’s FDI in the United States

I believe there this is not out-of-order. I am however, seeking an explanation to the equation. 

I have tried to check different sources and the closest explanation is that the function is a linear-log function.
This is not a structural paper. The authors are not "deriving" a log-linear relationship from a model, which they then proceed to estimate. They rather start by postulat the relationship: 


  Let $C^J_{it}$ be the number of citations made by the patent
  applications Japanese firm i filed in year t to the cumulated stock of
  bindigenousQ U.S.-invented patents granted as of year $t$. I can then
  write the log of $C^J_{it}$ as a simple log-linear function of several
  other observables
  
  $$ c^J_{it} = \beta_0 + \beta_1 p_{it} + \beta_2 FDI_{it} + \gamma_i +
&gt; \alpha_t + \epsilon_{it}$$


which is their equation (1).