对数标度

Log scale: intuition

Now that we're familiar with writing exponential equations, let's take this concept a step further!

We're used to seeing number lines spaced apart by integer increments; one space from $1$ to $2$ illustrates $1+1=2$, three spaces from $8$ to $11$ represents $3+8=11$, and so on. In general, our usual number line visually represents that $m$ spaces between the numbers $n$ and $p$ is an equivalent statement to $m+n=p$. If the spaces have length $l$ (instead of $1$), this becomes $lm+n=p$.

If we have data that covers a set of values that are all close together, the number line - or, linear scale - can handle it. Most graphs we initially learn about simply compare two linearly scaled pieces of information:

The line $y=x$ (also written $f(x)=x$ and called the identity function) takes in numbers and maps them to themselves.

We can see that both axes are number lines.

This graph of the unusual correlation (potential relationship) between cheese and civil engineering also uses a linear scale.

We can see here that the difference between any two points on any axis is the same along that axis.

Let's think about what happens when some information we want to look at covers a large range of values!

One possibility is the logarithmic scale; instead of incrementing values by $1$ (or by multiples of $1$), it uses the powers of a number. A number line with a log scale could use powers of $10$, for example, where a value would be $10^0$, then $10^1$, then $10^2$, and so on. Negative exponents of $10$ would be used to represent fractions.

On this new line, we see that a space between two values means that we multiply the earlier one by $10$ to get to the later one; three spaces between $10^2$ and $10^5$ means that $10^5$ is a thousand ($10^3$) times larger than $10^2$!

While $10$ is a common base for the log scale, any number would work. In the general case, $m$ spaces between $n$ and $p$ with a base $l$ is an equivalent statement to $l^m*n=p$.

This is extremely useful for graphing large data ranges. The decibel (dB), the measurement for sound, is actually logarithmic, which is why increasing sounds by a $40$ dB can be the difference between a normal conversation ($60$ dB) and a jackhammer ($100$ dB)!

In this graph, there are so many different values to be represented that they are split up by powers of $10$:

While this is not a typical graph, we can see the benefit of the log scale.

Log graphs can also be useful to find relationships between data that linear graphs could not communicate easily:

The plotted points remain unchanged; only the vertical axis is adjusted.

The above graphs show what happens when one axis uses a linear scale and the other a logarithmic one; there are, however, graphs where both axes are logarithmic (just as there are graphs where both axes are linear). This comparison of three functions where only the axes change demonstrates how the same data can be represented differently by changing the scale:

Check out the logarithmic graphing section here to learn how to graph these on your own!

(Video) Log Scale

by katari

Here's a good video by katari explaining what a logarithmic scale is and how it differs from a linear scale.

$\large{\text{Summary}}$

A linear scale increases in increments of 1 so it's additive. A logarithmic scale increases by a exponential increment of 1 or multiplication increments so it's multiplicative.

Logarithmic Scales

A logarithmic scale is a special type of number line used when numbers grow or shrink really fast. Instead of increasing by adding the same amount (like $1$, $2$, $3$), a log scale increases by multiplying (like $1$, $10$, $100$, $1000$).

Each step on a logarithmic scale represents a power of $10$, or sometimes a different base. For example, going from $10$ to $100$ on a log scale is the same size step as going from $100$ to $1,000$, even though the actual increase is much bigger.

We use log scales when numbers cover a huge range, and it would be too hard to fit them on a normal graph. They're used in science, engineering, and even music. Some common examples are:

• The Richter scale: for earthquakes
• The decibel scale: for sound
• The pH scale: for acids and bases

All of these are log scales, meaning each step is $10x$ bigger or smaller than the one before.

Log scales help you compare really large or really small values without needing giant graphs. They also make exponential growth look like a straight line, which is useful when analyzing data.

So, whenever you see strange scales like $1$, $10$, $100$, or $1,000$, chances are you’re looking at a logarithmic scale!

Looking Further

In the next few lessons, we'll continue to explore different rules and laws relating to log scales. The next lessons will cover Benford's Law and Euler's Number.