Identify the parent function (y=log_b(x))
Determine the base (b)
Find the vertical asymptote
Set the argument equal to zero to locate the asymptote
Determine the domain from the argument of the logarithm
Find the range, which is all real numbers
Plot the vertical asymptote as a dashed line
Choose easy x-values that make the logarithm simple
Compute corresponding y-values for those x-values
Plot key points such as ((1,0)), ((b,1)), and ((1/b,-1)) when applicable
Apply horizontal shifts by replacing (x) with (x-h)
Apply vertical shifts by adding (k)
Apply reflections if the function is multiplied by a negative
Apply vertical stretches or compressions if the function is multiplied by a constant
Draw the curve approaching the asymptote and passing through the key points
Check that the graph increases for bases greater than 1
Check that the graph decreases for bases between 0 and 1
Verify the graph stays on the correct side of the asymptote
Label the asymptote, intercepts, and key points if needed