Why logarithms are important

So that's what the logarithm function does. Why is that useful? Well, for the same reason that being able to undo an addition or a multiplication is useful. It lets you work backwards through a calculation. It lets you undo exponential effects. Beyond just being an inverse operation, logarithms have a few specific properties that are quite useful in their own right:. Logarithms are a convenient way to express large numbers. The base logarithm of a number is roughly the number of digits in that number, for example.

Slide rules work because adding and subtracting logarithms is equivalent to multiplication and division. This benefit is slightly less important today. Lots of things "decay logarithmically". For example, hot objects cool down, cold objects warm up. Things in motion experience friction and drag and gradually slow down. If you can take a problem and split it into two smaller problems that can be solved independently, you can probably write a computer program where the number of steps required to solve the problem is "logarithmic".

That is, the time taken depends on the logarithm of the amount of data to be processed. Logarithms are defined as the solutions to exponential equations and so are practically useful in any situation where one needs to solve such equations such as finding how long it will take for a population to double or for a bank balance to reach a given value with compound interest.

Historically, they were also useful because of the fact that the logarithm of a product is the sum of the logarithms and sums are easier to calculate by hand or to estimate by overlapping rulers as in a slide rule. In addition to providing a computational "trick", this property is the basis of the mapping property described in Christian Blatter's answer and generalizes to the concept of self-adjoint generators of unitary groups, which has many mathematical applications and relates physical observables to symmetry properties in quantum mechanics.

In some instances e. The logarithm provides a natural means to transform one view into the other: The sum of two shifts corresponds to the composition of two scalings.

Like many types of functions, the exponential function has an inverse. This inverse is called the logarithmic function, and it is the focus of this chapter. It describes how to evaluate logarithms and how to graph logarithmic functions. This section also addresses the domain and range of a logarithmic function, which are inverses of those of its corresponding exponential function. The next section presents two special logarithmic functions--the common logarithmic function and the natural logarithmic function.

This is a logarithmic scale, which in my head means "PageRank counts the number of digits in your score". So, a site with pagerank 2 "2 digits" is 10x more popular than a PageRank 1 site. How'd I do that? They might have a few times more than that M, M but probably not up to M. We're at the typical "logarithms in the real world" example: Richter scale and Decibel. The idea is to put events which can vary drastically earthquakes on a single scale with a small range typically 1 to Just like PageRank, each 1-point increase is a 10x improvement in power.

The largest human-recorded earthquake was 9. Decibels are similar, though it can be negative. Sounds can go from intensely quiet pindrop to extremely loud airplane and our brains can process it all.

In reality, the sound of an airplane's engine is millions billions, trillions of times more powerful than a pindrop, and it's inconvenient to have a scale that goes from 1 to a gazillion.

Logs keep everything on a reasonable scale. You'll often see items plotted on a "log scale". In my head, this means one side is counting "number of digits" or "number of multiplications", not the value itself. Again, this helps show wildly varying events on a single scale going from 1 to 10, not 1 to billions. Moore's law is a great example: we double the number of transistors every 18 months image courtesy Wikipedia.

The neat thing about log-scale graphs is exponential changes processor speed appear as a straight line. Growing 10x per year means you're steadily marching up the "digits" scale.

If a concept is well-known but not well-loved, it means we need to build our intuition. Find the analogies that work, and don't settle for the slop a textbook will trot out. In my head:. Learn Right, Not Rote. The number 1,, is , times as big as 10, but it only has five more digits. The number of digits a number has grows logarithmically. And thinking about numbers also shows why logarithms can be useful for displaying data. Can you imagine if every time you wrote the number 1,, you had to write down a million tally marks?

Log scales can be useful because some types of human perception are logarithmic. In the case of sound, we perceive a conversation in a noisy room 60 dB to be just a bit louder than a conversation in a quiet room 50 dB.

Yet the sound pressure level of voices in the noisy room might be 10 times higher. Another reason to use a log scale is that it allows scientists to show data easily. It would be hard to fit the 10 million lines on a sheet of graph paper that would be needed to plot the differences from a quiet whisper 30 decibels to the sound of a jackhammer decibels.

Logarithms have many uses in science. So is the Richter scale for measuring earthquake strength.