Welcome to logarithmic graphing! A logarithmic function is the inverse of an exponential function.
The basic form is y equals log base b of x. Here we see the graph of y equals log base 2 of x.
Notice the vertical asymptote at x equals zero, and key points like one comma zero, two comma one, and four comma two.
To find the vertical asymptote of a logarithmic function, we need to identify where the argument becomes zero.
For y equals log base 2 of x minus 2, we set x minus 2 greater than zero.
The boundary occurs when x minus 2 equals zero, so x equals 2 is our vertical asymptote.
The domain is all x values greater than 2, shown by the green arrow.
To find key points for logarithmic functions, we use special values of the argument.
For y equals log base 2 of x minus 2, when the argument x minus 2 equals 1, we get x equals 3 and y equals 0.
When the argument equals the base 2, we get x equals 4 and y equals 1.
When the argument equals one half, we get x equals 2.5 and y equals negative 1.
These key points help us sketch the curve accurately.
Logarithmic functions can be transformed in several ways.
The blue curve shows the basic function y equals log base 2 of x.
The red curve shows y equals log base 2 of x minus 2 plus 1, which shifts the graph right 2 units and up 1 unit.
The green curve shows y equals negative log base 2 of x, which reflects the graph over the x-axis.
Notice how the vertical asymptotes also shift with horizontal transformations.
Let's complete the graphing process with an example: y equals 2 log base 3 of x plus 1 minus 1.
First, we find the vertical asymptote at x equals negative 1. The domain is x greater than negative 1.
We plot key points like zero comma negative 1, two comma 1, and eight comma 3.
Finally, we sketch a smooth curve that approaches the asymptote but never touches it.
This systematic approach works for any logarithmic function!