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"

  • When a=1, the graph is not defined
  • Apart from that there are two cases to look at:

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:

  • It is always on the positive side of (and never crosses) the y-axis
  • It always intersects the x-axis at x=1 ... in other words it passes through (1,0)
  • At x=a, f(x)=1 ... in other words it passes through (a,1)
  • It 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 (and more ...)

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 remeber 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.