Ln(l -I- Prod) is the

natural logarithm of one plus Prod and is our innovation output measures.

In fact, the

natural logarithm of the electron-to-proton wavelength ratio is approximately 7.5 and consequently, F calibrated on the proton will be shifted by 7.5 logarithmic units relative to the F calibrated on the electron:

Scatterplots of the dependent variable LN_TBCOEF10 related to the independent variables LN_PEROOM, LN_ILLITER, and LN_INADSAN (the variables transformed to

natural logarithm now have the prefix "LN_") were then constructed.

Regression of the

natural logarithm of raw counts on time yielded a naive estimate of the intrinsic growth rate (r) of 0.084 (SE = 0.019), suggesting an average annual discrete growth rate ([lambda]) of ~1.09.

More specifically, [y.sub.t] = ([food.sup.*.sub.t] [gdp.sub.t] [[pi].sub.t] [i.sub.t] [e.sub.t])', where [food.sup.*.sub.dt is the

natural logarithm of the international food price index for a specific country, [gdp.sub.t] is the

natural logarithm of that country's gdp, Pt is the

natural logarithm of the country's headline inflation index, [i.sub.t] is an interest rate closely following the central bank's policy rate, and e t is the

natural logarithm of that country's nominal exchange rate measured as units of local currency per unit of US Dollars.

(5) When variables are expressed in

natural logarithm this was taken over the Tornqvist quantity index

Thus, our empirical specifications include the

natural logarithm of REER to capture the influence of relative prices.

It was found that only their respective

natural logarithms underwent a normal behavior (P>0.40).

In all cases, the basic idea is that the

natural logarithms of variances are modeled using a linear model to account for heterogeneity of the variances (on a logarithmic scale) in terms of covariates and factor levels.

Mathematical descriptions of certain biological transformations in soil are best described by using

natural logarithms. Most calculators have the capacity to compute

natural logarithms directly.

To document growth, the data were transformed to

natural logarithms (7).

To do so, we took the

natural logarithms of all the rates available to us and then regressed the

natural logarithms of the wages, fatality rates, and injury and illness rates against the year of observation.