# Dimensions in the logarithmic form of the Arrhenius equation [duplicate]

The logarithmic form of the Arrhenius equation is:

$$\displaystyle\ln k=\ln A-\frac{E_a}{RT}$$

Here $$k$$ and $$A$$ have dimensions whereas $$\displaystyle\frac{E_a}{RT}$$ is dimensionless. In other words, $$k$$ and $$A$$ have dimensions whereas $$\ln k$$ and $$\ln A$$ are dimensionless. How can this be?

## 1 Answer

As pointed out by @G. Smith in the comments, your logarithms are all dimensionless function of dimensionless quantities.
The initial equation is in fact consistent only if $$k$$ and $$A$$ share the same unit (which is, in this case, $$s^{-1}$$); they can thus both be rewritten as $$k = \bar k u$$ and $$A = \bar A u$$, where $$u = 1 s^{-1}$$ and $$\bar k$$,$$\bar A$$ are dimensionless quantities.
Therefore, using the properties of logarithms, $$\log k - \log A = \log \bar k - \log \bar A$$ and your logarithms are dimensionless.