Logarithmic Function Reference

This is the Logarithmic Function:

f(x) = log_a(x)

a is any value greater than 0, except 1

a between 0 and 1		a above 1

Example: f(x) = log_½(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

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:

Inverse

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.

This is the "Natural" Logarithm Function:

f(x) = log_e(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: log_e(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.

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

log_a(x) = ln(x)ln(a)

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