How to interpret log differences in a partial log-log regression

I'm currently trying to understand the relationship between firm performance and various independent variables (e.g. firm size, firm profits..). Now, the regression I'm estimating looks like the following:

Δlog(firm_performance) = α + β1 Δlog(firm_size) + β2(other_variable) + ε

Where Δ represents the difference between the observation at time t and the observation at time t-1 and the logarithm of other_variable has not been taken.
How do I correctly read and understand the results if "other_variable" is found to be positively and significantly correlated with "firm_performance"? Does an increase in "other_variable" cause an increase in "firm_performance"? Or does an increase in "other_variable" cause an increase in the variance of "firm_performance" (Δfirm_performance)?
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Active Member
'sup. im going with, an increase in other_variable gives an exp(B2) fold increase in the geometric mean fold increase, year on year. so its a ratio of ratios type thing. i could be wrong about that but test it out.