Logarithm
A logarithm is the inverse function to an exponent.
- the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x.
- ex. if b = 10 then we must raise it to the 3rd power to get 1000. "The logarithm base 10 of 1,000 is 3
is said "3 is the logarithm of 8 to base 2"
Another way of saying "what is the logarithm?" is "to what power do I have to raise 10 in order to get a result in 100?"
Logarithms can be converted between any positive bases except 1, since all of its powers are equal to 1
Logarithmic scale
If a normal scale increases by 10 each evenly spaced notch, then a logarithmic scale would increase by 10x each evenly spaced notch
- regular - 10, 20, 30, 40...
- logarithmic - 10, 100, 1000, 10000...
- Thus moving a unit of distance along the scale means the number has been multiplied by 10
- note: not always 10, but always some fixed factor
- Thus moving a unit of distance along the scale means the number has been multiplied by 10
This allows us to display numerical data over a very wide range of values in a compact way
- typically the largest numbers in the data are hundreds or even thousands of times larger than the smallest numbers
Application
Logarithms are useful for solving equations in which the unknown appears as the exponent of some other quantity. For example, logarithms are used to solve for the half-life, decay constant, or unknown time in exponential decay problems. They are important in many branches of mathematics and scientific disciplines, and are used in finance to solve problems involving compound interest.
Finding result of 2 numbers multiplied together
100 × 1,000 can be calculated by looking up the logarithms of 100 (2) and 1,000 (3), adding the logarithms together (5), and then finding its antilogarithm (100,000) in the table.