# What is a logarithmic model?

Sep 17, 2015

A logarithmic model is a model that measures the magnitude of the thing it's measuring. It can also be seen as the inverse of an exponential model.

#### Explanation:

For example, exponential growth is very common in nature for things like radioactivity, bacterial growth, etc., being written as

$N \left(t\right) = {N}_{0} {e}^{k t}$ or $N \left(t\right) = {N}_{0} {a}^{t}$

So if wanted to know how much time passed based on the amount there is, we'd have a logarithmic model.

$\ln \frac{\frac{N \left(t\right)}{N} _ 0}{k} = t$ or ${\log}_{a} \left[\frac{N \left(t\right)}{N} _ 0\right] = t$

However, the bigger use of logarithmic models are when the thing you're measuring can go from things being very small to very big. For example, two common scales used are the Bell scale for sound intensity and the pH scale for acidity.

These values, before undergoing the logarithmic treatment usually range from ${10}^{-} 12$ to ${10}^{8}$ and ${10}^{-} 14$ to $1$ respectively. When we have such a big range we usually apply logarithms because then we have less powers, multiplications, etc. to work with.

With logs, the values range from $0 \mathrm{db} \eta$ to $200 \mathrm{db} \eta$ and from $1$ to $14$, which are much more palatable.

Logarithmic models often have these formulas

$y = \log \left(\frac{x}{x} _ 0\right)$ or $y = \log \left({x}_{0} / x\right)$

Where ${x}_{0}$ is a reference value. Whether it's on the numerator or the denominator depends on the range. If, like on the pH scale we're working with values much too small, it's easier to put it on top, since the biggest feasible number will be $1$.

But since we can work with values up to ${10}^{8}$ in the Bell Scale it's easier for it to be on the bottom. For reference these are the values for these two examples:

$\beta = \log \left(\frac{I}{10} ^ \left(- 12\right)\right)$ and
$p H = - \log \left[{H}^{+}\right] = \log \left(\frac{1}{\left[{H}^{+}\right]}\right)$

And last but not least, anything that's p something, is usually a logarithmic scale.