Welcome to logarithmic graphing! A logarithmic function has the form y equals log base b of x, where b is the base and x must be positive. The key features include a domain of x greater than zero, a vertical asymptote at x equals zero, and the graph always passes through the point one comma zero. When the base is greater than one, the function is increasing.
Logarithmic functions can be transformed using the general form y equals a times log base b of x minus h plus k. The parameter h creates a horizontal shift, moving the vertical asymptote from x equals zero to x equals h. The parameter k creates a vertical shift, and a creates vertical stretching or compression. Here we see the base function in blue and a transformed version in red, shifted right by 2 units and up by 1 unit.
To graph a logarithmic function accurately, we need to find key points. First, identify the vertical asymptote by setting the argument equal to zero. For y equals log of x minus 1 plus 1, the asymptote is at x equals 1. Next, find the x-intercept by setting y equal to zero. Then choose additional points like x equals 2 and x equals 1 plus e to help sketch the curve. These key points guide us in drawing the complete graph.
The base of a logarithmic function significantly affects its shape. When the base is greater than 1, the function is increasing, with steeper curves for larger bases. Common bases include base 10 for common logarithms, base e for natural logarithms, and base 2 for binary logarithms. All logarithmic functions pass through the point one comma zero and have the same general shape, but with different steepness depending on the base.
Let's work through a complete example: graphing y equals 2 log base 3 of x plus 1 minus 1. First, identify the transformations: h equals negative 1 shifts the graph left 1 unit, k equals negative 1 shifts it down 1 unit, and a equals 2 stretches it vertically by factor 2. The vertical asymptote moves to x equals negative 1. We find key points like zero comma negative 1 and 2 comma 1, then sketch the curve through these points, approaching the asymptote.