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 isn't defined
  • Apart from that there are two cases to look at:

a between 0 and 1

a above 1
Graph of log base a of x where a is between 0 and 1, showing a curve falling from left to right. Graph of log base a of x where a is greater than 1, showing a curve rising from left to right.
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)

Play with the graph here. The "a" value is the base. (Why ln(x)/ln(a)? see below.)

../algebra/images/function-graph.js?fn0=ln(x)/ln(a)&xmin=-6&xmax=16&ymin=-8&ymax=8&vara=2|0|10

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)

See them both together here: ../algebra/images/function-graph.js?fn0=ln(x)/ln(a)&fn1=a^x&xmin=-6&xmax=16&ymin=-8&ymax=8&vara=2|0|10

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).

Graph of the natural logarithm ln(x) passing through (1,0) and (e,1). 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.

In the graph above, why did we use ln(x)/ln(a)?

To graph a log with any base a we only need the Natural Logarithm function ln and the Change of Base Formula:

loga(x) = ln(x)ln(a)

This allows the slider to change the base a dynamically, recalculating the curve in real-time.

Common Functions Reference Algebra Index

Tag » How To Graph Logarithmic Functions