Logarithmic Function Reference

This is the Logarithmic Function:

f(x) = loga(x)

a is any value greater than 0, except 1

Properties depend on value of "a"

a between 0 and 1


a above 1

logarithm function   logarithm function
Example: f(x) = log½(x)
Example: f(x) = log2(x)

For a between 0 and 1


For a above 1:

Plot the graph here (use the "a" slider)

In general, the logarithmic function:

  • always has positive x, and never crosses the y-axis
  • always intersects the x-axis at x=1 ... in other words it passes through (1,0)
  • equals 1 when x=a, in other words it passes through (a,1)
  • is an Injective (one-to-one) function

Its Domain is the Positive Real Numbers: (0, +∞)

Its Range is the Real Numbers: Real Numbers


loga(x)   is the Inverse Function of  ax (the Exponential Function)

So the Logarithmic Function can be "reversed" by the Exponential Function.

The Natural Logarithm Function

This is the "Natural" Logarithm Function:

f(x) = loge(x)

Where e is "Eulers Number" = 2.718281828459... etc

But it is more common to write it this way:

f(x) = ln(x)

"ln" meaning "log, natural"

So when you see ln(x), just remember it is the logarithmic function with base e: loge(x).


natural logarithm function
Graph of f(x) = ln(x)

At the point (e,1) the slope of the line is 1/e and the line is tangent to the curve.