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
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, 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:
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... 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: loge(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.