# what exactly *is* a logarithm and what does it do?

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I mean, I’ve used them in algebra many times but I never really understood what it does. Kinda like in biostatistics how I could do the math, but how it worked was beyond me entirely.

So yeah: like what’s this sorcery and what does it do/why do we use it?

In: Mathematics

Division is the opposite of multiplication. The result of a/b is “the number that you multiply by b to get a”.  If a/b = c, then b * c = a

Logarithms are the same things for exponents. Log_b(a) is “the power that you raise b to in order to equal a”. If log_b(a) = c, then b^c = a

Subtraction is the inverse of addition.

Division is the inverse of multiplication.

Logarithms are the inverse of exponents.

Using logarithms to plot data can help us see when exponential growth or decay are happening because that will look linear on the plot and linear is easy to recognize. It also helps because a 10% jump looks the same whether it’s from 1 to 1.1 or from 1,000 to 1,100. If you care about relative changes, logarithmic plotting helps you recognize that.

It gives the answer to the question “How many times a number is multiplied to get the other number?”. It is a mathematic equation that makes it easier to represent trends in data that might be increasing exponentially.

For example if you had something that was increasing exponentially: bacteria reproduction or compounding interest, for example, where the numbers would double every few periods, it quickly becomes not very informative to show the increase in a linear fashion as the scaling becomes really a trend/curve. So instead you can plot it by using the Log of the number.

A table of logarithms is what you had to use before calculators to take roots and exponents. You can take a logarithm of a number divide it by 3 if you wanted the third root then look up the inverse. Also some physics and biology equations follow a logarithmic curve

It’s a reverse-exponent. You want it when you to know what the exponent was.

A great example is radioactive decay. We know that certain substances, like Carbon-14, reduce by half in a certain time frame. Although that amount is getting smaller instead of bigger, it is still an exponential function: 1/2 ^ (k*t). (k is a known constant, and t is the variable time.)

But you don’t want to calculate the amount remaining – you know what. You want to calculate how much time passed, which is that exponent. Boom: logarithm!

So any time you want to get *the exponent itself*, and whatever else that might tell you, you use a logarithm.

Logarithms feel a bit like asking “how many digits (after the first) does this number have?” or “what order of magntidue is this number?”, although with some more nuance.

Consider log to the base 10.

So log1=0

log10=1

log100=2

log1000=3

etc, so for any power of 10, log(base10) gives us that power.

The log function also gives us an answer in-between these values, like log50 ~= about 1.7.

Often people will use base-2 to talk about computing. This will sometimes imply some logarithms, telling us how many *binary* digits (1s and 0s) the computer is using.

For instance “This is a 64 bit processor.” means “The log (to base 2) of the number of memory addresses this processor can imagine, is 64.” i.e. “It can imagine 2^64 memory addresses.”

Mathematicians often use log to base “e”, where e refers to a specifical useful constant, similar to how pi is useful, in how it appears in many interesting formula.

Logarithm, the exponent or power to which a base must be raised to yield a given number. Expressed mathematically, x is the logarithm of n to the base b if bx = n, in which case one writes x = logb n. For example, 23 = 8; therefore, 3 is the logarithm of 8 to base 2, or 3 = log2

A logarithm is like asking, “To what power do we need to raise this base number to get another number?” For example, if we ask, “What power do we raise 10 to get 100?” The answer is 2 because 10^2=100. We use logarithms to simplify complex multiplications and to solve equations involving exponential growth, like in biology, finance, and computer science. It’s a way to make big numbers easier to work with and understand.

I’d just add to these explanations that it’s often helpful to look at some data after taking a logarithm. (Most of the time you take the log and multiply by 10). When you graph data that lives on a really big scale. This is clear from the fact that a log operation turns 100 into 2 and 1000 into 3, using the 10x method,, 100 becomes 20 and 1000 becomes 30. When you do it this way, the units of the answer are usually said to be in decibels which is written dB for short.

One thing commonly measured in dB is the “loudness” of sound as perceived by the ear. So if the sound goes up by 10 dB, the underlying data is going up by a factor of 10. 20 dB is a factor of 100.

The threshold of pain is up around 80 or 90 dB, and quiet is around 10 or so. The ear is very sensitive over a huge range!

Earthquakes are measured on the Richter scale, which doesn’t use the 10x, which means going up by 1, makes the earthquake 10x more powerful.

It’s also very common to use in engineering when comparing things like signal power and noise power, and many other comparative measures.