Logarithmic Function Reference - Math Is Fun

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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

  • As x nears 0, it heads to infinity
  • As x increases it heads to -infinity
  • It is a Strictly Decreasing function
  • It has a Vertical Asymptote along the y-axis (x=0)

For a above 1:

  • As x nears 0, it heads to -infinity
  • As x increases it heads to infinity
  • it is a Strictly Increasing function
  • It has a Vertical Asymptote along the y-axis (x=0)

Plot the graph here

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

Inverse

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 so on

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.

Common Functions Reference Algebra Index

Tag » How To Graph Logarithmic Functions