Logarithmic scale

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A logarithmic scale is a scale of measurement that uses the logarithm of a physical quantity instead of the quantity itself. Presentation of data on a logarithmic scale can be helpful when the data covers a large range of values; the logarithm reduces this to a more manageable range. Some of our senses operate in a logarithmic fashion (doubling the input strength adds a constant to the subjective signal strength), which makes logarithmic scales for these input quantities especially appropriate. In particular our sense of hearing perceives equal ratios of frequencies as equal differences in pitch.

Logarithmic scales are either defined for ratios of the underlying quantity, or one has to agree to measure the quantity in fixed units. Deviating from these units means that the logarithmic measure will change by an additive constant. The base of the logarithm also has to be specified, unless the scale's value is considered to be a dimensional quantity expressed in generic (indefinite-base) logarithmic units.

On most logarithmic scales, small values (or ratios) of the underlying quantity correspond to small (possibly negative) values of the logarithmic measure. Well-known examples of such scales are:

• Richter magnitude scale for strength of earthquakes and movement in the earth.
• bel and decibel and neper for acoustic power (loudness) and electric power;
• cent, minor second, major second, and octave for the relative pitch of notes in music;
• logit for odds in statistics;
• Palermo Technical Impact Hazard Scale;
• Logarithmic timeline;
• counting f-stops for ratios of photographic exposure;
• rating low probabilities by the number of 'nines' in the decimal expansion of the probability of their not happening: for example, a system which will fail with a probability of 10^-5 is 99.999% reliable: "five nines".
• Entropy in thermodynamics.
• Information in information theory.

Some logarithmic scales were designed such that large values (or ratios) of the underlying quantity correspond to small values of the logarithmic measure. Examples of such scales are:

• pH for acidity;
• stellar magnitude scale for brightness of stars;
• Krumbein scale for grain size in geology.
• Kardashev scale for technological advance in physics.

[edit] Graphic representation

A logarithmic scale is also a graphic scale on one or both sides of a graph where a number x is printed at a distance c·log(x) from the point marked with the number 1. A slide rule has logarithmic scales, and nomograms often employ logarithmic scales. On a logarithmic scale an equal difference in order of magnitude is represented by an equal distance. The geometric mean of two numbers is midway between the numbers.

Logarithmic graph paper, before the advent of computer graphics, was a basic scientific tool. Plots on paper with one log scale can show up exponential laws, and on log-log paper power laws, as straight lines (see semilog graph, log-log graph).

[edit] Estimating values in a diagram with logarithmic scale

To estimate the value of a point on a logarithmic axis:

Now this might be a tricky task to do if you want to get a value that doesn't differ too much from reality. For some uses an educated guess simply can't give you decent enough error margins (i.e. engineering purposes). The easy way to get a more correct value is the following:

1. Measure the distance from the point on your scale to the nearest decade line with lower value.
2. Divide this distance by the length of a decade. (the length between two decade lines)
3. The value of your chosen point is now the value of the nearest decade line with lower value times 10^a where a is the value you found in step 2.

Example: What is the value that lies halfway between the 10 and 100 decades on a logarithmic axis? Since we are interested in the halfway point, the quotient of steps 1 and 2 is 0.5. The nearest decade line with lower value is 10, so the halfway point's value is (10^0.5)*10.

To estimate where a value lies within a decade on a logarithmic axis, use the following method:

1. Measure the distance between consecutive decades with a ruler (english or metric). You just need to be consistent with your intervals.
2. Take the log(value of interest/nearest lower value decade) multiplied by the number you determined in step one.
3. Using the same units as in step 1, count as many units as resulted from step 2 - starting at the lower decade.

Example: Let's say you want to know where 17 is located on a logarithmic axis. Use your ruler to measure the distance between 10 and 100. Let's say number of intervals is 30 on my ruler (note it can vary - you just need to be using the same scale through the rest of the process).

You take the log (17/10) * 30 = 6.9

So, on the interval 7 of the 30 intervals I counted across the decade, you will find that is where 17 belongs.

[edit] See also

• Logarithmic units