Identify the logarithmic function, such as y = log_b(x), y = ln(x), or y = a log_b(x – h) + k
Determine the base b and any transformations
Find the vertical asymptote
For y = log_b(x), the vertical asymptote is x = 0
For y = log_b(x – h), the vertical asymptote is x = h
Plot key points using known log values
For y = log_b(x), use points such as (1, 0), (b, 1), and (1/b, -1)
Apply shifts, stretches, and reflections to the key points if needed
Draw the asymptote as a dashed line
Plot the transformed key points
Sketch a smooth curve through the points
Make the curve approach the asymptote without crossing it
Ensure the graph exists only on the domain where the log input is positive
Determine the domain from the inside of the logarithm
For y = log_b(x), domain is x > 0
For y = log_b(x – h), domain is x > h
Determine the range
For logarithmic functions, the range is all real numbers
Check intercepts if needed
Find the x-intercept by setting y = 0
Find the y-intercept only if the domain includes x = 0
Use a table of values if more points are needed
Verify the graph matches the base and transformations
Label axes and important points clearly