Logarithmic Function Reference
This is the Logarithmic Function:
f(x) = log_{a}(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 

Example: f(x) = log_{½}(x) 
Example: f(x) = log_{2}(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 yaxis
 always intersects the xaxis 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 (onetoone) function
Its Domain is the Positive Real Numbers: (0, +∞)
Its Range is the Real Numbers:
Inverse
log_{a}(x) is the Inverse Function of a^{x} (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) = log_{e}(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: log_{e}(x).
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.