Logarithmic functions are the inverse of exponential functions. The basic form is y equals log base b of x. Key features include a vertical asymptote at x equals zero, the domain is all positive real numbers, and the graph passes through the point one comma zero for any base.
The first step in graphing logarithmic functions is to identify the function form. The general form is y equals a times log base b of quantity x minus h plus k. Here, a represents vertical stretch or compression, b is the base, h is the horizontal shift, and k is the vertical shift. Compare the basic function y equals log base 2 of x with the transformed function y equals 2 log base 2 of quantity x minus 1 plus 1.
Step two is finding the vertical asymptote. For a logarithmic function y equals log base b of quantity x minus h, set the argument equal to zero. This gives us x minus h equals zero, so x equals h. The vertical asymptote is the line x equals h. For example, with y equals log base 2 of quantity x minus 3, the asymptote is at x equals 3. The domain is all x values greater than h, since the argument must be positive.
Step three is finding key points to plot the graph accurately. First, find the x-intercept by setting y equal to zero and solving for x. For additional points, choose x values that make the argument equal to simple powers of the base. For y equals log base 2 of quantity x minus 3, when x minus 3 equals 1, we get x equals 4 and y equals 0. When x minus 3 equals 2, we get x equals 5 and y equals 1. When x minus 3 equals 4, we get x equals 7 and y equals 2. These key points help us sketch the curve.
The final step is sketching the complete graph by connecting all the elements. Draw a smooth curve that approaches the vertical asymptote but never touches it, passes through all the key points, and extends toward infinity. The logarithmic function always increases from left to right when the base is greater than one. Remember the domain is all x values greater than h, and the range is all real numbers. Compare the basic function y equals log base 2 of x with the shifted function y equals log base 2 of quantity x minus 3 to see how transformations affect the graph.