Calculus How To

Logarithm Function: Definition

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Types of Functions >

You might want to read this article first: What is a Logarithm?

What is a Logarithm Function?

A logarithm function is a function of the form:

f(x) = logbx.

It is the inverse function of the exponential function. In notation, that’s: If y = logbx, then by = x.

logarithm function

The logarithmic function logbx is read “log base b of x”.

Sometimes a logarithm function is just written as log x. In those cases the writer thought which base to use was obvious, and we use our knowledge of the situation to determine what the expression means exactly. In the vast majority of cases, if you see “log x”, then assume it’s base 10.

Logarithm Function Properties

A knowledge of the basic properties of logarithmic functions may help you in your evaluations.

  • logb1 = 0 . Since b0 = 1, the exponent any base needs to make 1 is always zero.
  • logbb = 0 . Since b1 = b, the exponent any base needs to make itself is always just one.
  • logbbx = x . Remember log bas b of x and b to the x power are inverse functions. When inverse functions are applied to each other, the inverse out (essentially cancel each other out).
  • If we know that logbx = logby, then we know that x = y.
  • If we know that logbx = logax, then we know that a = b.

Special Bases

The logarithm with base 2 is called the binary logarithm, and it is foundational in much of computer science.

The logarithm with base e (≈ 2.718) is called the natural logarithm and is important in math and physics.

The logarithm with base 10 is the common logarithm, and is used in science, engineering and many other fields.


Jones, James. Logarithmic Functions and Their Graphs. Retrieved from on March 13, 2019

Stephanie Glen. "Logarithm Function: Definition" From Calculus for the rest of us!

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