An example is the decibel scale, which measures sound pressure level. It is one way to describe the strength of a sound wave. For every 10 decibel increase, the sound pressure increases 10 times. So a 20 decibel sound has not twice the sound pressure of 10 decibels, but 10 times that level. An exponent is a number that tells you how many times to multiply some base number by itself. In that example above, the base is So using exponents, you could say that 50 decibels is 10 4 times as loud as 10 decibels.
Exponents are shown as a superscript — a little number to the upper right of the base number. Logarithms are the inverse of exponents.
For instance, how many times must a base of 10 be multiplied by itself to get 1,? So the logarithm base 10 of 1, is 3. At first, the idea of a logarithm might seem unfamiliar. But you probably already think logarithmically about numbers. The number is 10 times as big as the number 10, but it only has one more digit. The number 1,, is , times as big as 10, but it only has five more digits. It only takes a minute to sign up.
Connect and share knowledge within a single location that is structured and easy to search. You can "undo" addition by performing subtraction. You can "undo" multiplication by performing division. 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.
For example pH the number of hydrogen atoms present is too large up to 10 digits. To allow easier representation of these numbers, logarithms are used. 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.
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Logarithms are everywhere. Ever use the following phrases? Mama mia! Ok, ok, we get it: what are logarithms about? Logarithms find the cause for an effect, i. By the way, the notion of "cause and effect" is nuanced. Why is bigger than ?
Logarithms put numbers on a human-friendly scale. Logarithms count multiplication as steps Logarithms describe changes in terms of multiplication: in the examples above, each step is 10x bigger. Show me the math Time for the meat: let's see where logarithms show up! Six-figure salary or 2-digit expense We're describing numbers in terms of their digits, i.
Try it out here: Order of magnitude We geeks love this phrase. Interest Rates How do we figure out growth rates? My two favorite interpretations of the natural logarithm ln x , i. Google conveys a lot of information with a very rough scale Measurement Scale: Richter, Decibel, etc.
Logarithmic Graphs You'll often see items plotted on a "log scale". Onward and upward If a concept is well-known but not well-loved, it means we need to build our intuition. In my head: Logarithms find the root cause for an effect see growth, find interest rate They help count multiplications or digits, with the bonus of partial counts k is a 6. Continuous Growth What does an exponent really mean? Q: Why is e special? Join k Monthly Readers Enjoy the article?
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