What is an exponent?

The exponent of a number says how many times to use the number in multiplication

Example:

3^3=3x3x3=27 (3 is used 3 times in a multiplication to get 27)

What is Logarithm?

A logarithm goes the other way.

It asks the question "what exponent produced this?"

And the answer is:

The Natural Logarithm and Natural Exponential Functions

When the base is e("Euler's Number"=2,718281828459...) we get:

The Natural Logarithm loge(x) which is more commonly written In(x)

The Natural Exponential Function e^x

And the same idea that one can "undo" the other is still true:

In(e^x)=x

e^(In*x)=x

The Common Logarithm

When the base is 10 you get:

The Common Logarithm log10(x), which is sometimes written as log(x)

Engineers love to use it, but it is not used much in mathematics.

Example:

Calculate log10 100

Well, 10x10=100, so when 10 is used 2 times in multiplication you get 100:

LOG10100=2

Changing the Base

What if we want to change the base of a logarithm?

Follow this formula:

x goes up, a goes down

Or another way of thinking it is that logb a is like a "conversion factor"

LOGax= LOGbx/LOGba

Applications of exponential fucntions

Perhaps the most well-known application of exponential functions comes from the financial world.