Logarithmic Models

Key Questions

There are two main advantages: linearization and ease of computation / comparation, the former of which ties into the second.

Explanation:

The easier one to explain is the ease of computation / comparation.

The logarithmic system I think that's simple to explain is the pH model, which most people are at least vaguely aware, you see, the p in pH is actually a mathematical code for "minus log of", so pH is actually $- \log \left[H\right]$

And this is useful because in water, the H, or concentration of free protons (the more around, the more acidic), usually varies between $1 M$ and ${10}^{-} 14 M$, where $M$ is shorthand for mol/L, the appropriate unit of measurement, and yet, if we take the log the scale goes from $0$ to $- 14$, ( since we like to work with positive numbers we multiply by minus one, but that's besides the point)

Even though we lost the basic intuition we had with the original scale (where we know that, for example $1 M$ is twice more acidic than $0.5 M$) we're now working with a range easier to work with, not to mention that at least this particular systems works because usually we don't need the intuition we lost while doing this.

And that also helps with the first part, because you see, sometimes stuff in nature works exponentially, like for example, one type of analysis you might find in a chemical laboratory would look like this with raw data:

graph{10^(-x+2)+2 [-0.21, 19.79, -0.12, 9.88]}

But as soon as you take the log of it, it comes out more like

graph{x-2 [-0.21, 19.79, -0.12, 9.88]}

And the thing is, we can and like working with lines much more than that other curve, the line can be more easily manipulated, you can interpolate data much more easily, it's just simpler for the poor researchers to take the log.

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.

It depends on what you're doing.

Explanation:

Well, logarithms were historically used and created because they make products into sums and powers into multiplications.

So whenever you're working with an expression that involves a lot of products and powers, but not to many sums, it might be easier to take the log. For example

$y = 4 \left(\cos \left(2 x\right) + 5\right) \tan \left(x\right) {\sec}^{4} \left(x\right)$

This will likely involve the multiplication and the powers of a lot of messy numbers, if computing by hand, but, by taking the log, we can get

$\ln \left(y\right) = \ln \left(4\right) + \ln \left(\cos \left(2 x\right) + 5\right) + \ln \left(\tan \left(x\right)\right) + 4 \ln \left(\sec \left(x\right)\right)$

So then you can just look up the values on logarithm tables and do much easier sums.

It's very important in calculus too for precisely the same reason, it turns that differentiating sums is easier than differentiating products (and you'll be doing a lot of differentiating) and it also turns that the natural log has a convenient derivative too.

You can also use the logarithms to solve exponential equations of bases that can't be equal, for example, let $a$ and $b$ be two positive, non-zero, non-one numbers that aren't multiples or submultiples of each other, and let $f \left(x\right)$ and $g \left(x\right)$ be any two functions. So for the equation

${a}^{f \left(x\right)} = {b}^{g \left(x\right)}$

Just apply the logarithm

$f \left(x\right) \cdot \ln \left(a\right) = g \left(x\right) \cdot \ln \left(b\right)$

And attempt to isolate the x.

And, last but not least, to solve a logarithm equation, of the type
${\log}_{a} \left(f \left(x\right)\right) = k$

Just take the a-base expoential and solve it, i.e.:
${\log}_{a} \left(f \left(x\right)\right) = k \rightarrow f \left(x\right) = {a}^{k}$