How to Graph Logarithmic Functions

A logarithmic function has the form \( y = \log_b(x) \), where \( b \) is a positive number not equal to 1. Because logarithms and exponents undo each other, graphing a log function is really about understanding its inverse relationship with an exponential function. If you're not yet comfortable moving between the two forms, it helps to review how to convert from logarithmic to exponential form before working through graphs.

Every basic log graph shares a few features, no matter what base \( b \) is used:

• The domain is \( x > 0 \), so the curve only exists to the right of the y-axis.
• There is a vertical asymptote along the y-axis (\( x = 0 \)) that the curve approaches but never touches.
• The graph always passes through \( (1, 0) \), since \( \log_b(1) = 0 \) for any base.
• The graph always passes through \( (b, 1) \), since \( \log_b(b) = 1 \).

Take \( y = \log_2(x) \) as a starting example. Since the base \( 2 \) is greater than 1, the function is increasing, but it climbs more and more slowly as \( x \) grows. Plotting a handful of points makes this clear: \( (1, 0) \), \( (2, 1) \), and \( (4, 2) \) all satisfy the equation, and the curve hugs the y-axis as \( x \) approaches 0 from the right.

If the base were between 0 and 1 instead, the same shape would flip vertically: the curve would decrease from left to right instead of increasing, but it would still pass through \( (1, 0) \) and still have the y-axis as its asymptote.

The natural logarithm, written \( \ln(x) \), is just \( \log_e(x) \), where \( e \approx 2.718 \). Every rule that applies to \( \log_b(x) \) applies to \( \ln(x) \) as well: the domain is \( x > 0 \), the vertical asymptote is \( x = 0 \), and the curve passes through \( (1, 0) \). Because the base \( e \) is greater than 1, the graph of \( y = \ln(x) \) is increasing, and it passes through \( (e, 1) \) instead of \( (2, 1) \).

Students often ask how the log curve compares to values found without a calculator; practicing that skill in evaluating logarithms without a calculator makes it much faster to plot accurate points by hand.

Use this process for any function in the form \( y = \log_b(x - h) + k \):

1. Find the domain. Set the expression inside the log greater than 0 and solve. This tells you where the curve exists.
2. Locate the vertical asymptote. It occurs where the inside of the log equals 0. For \( y = \log_b(x - h) + k \), the asymptote is the vertical line \( x = h \).
3. Plot the key point. The base point \( (1, 0) \) on the parent graph shifts to \( (h + 1, k) \) after a horizontal shift of \( h \) and a vertical shift of \( k \).
4. Plot a second point. Use \( (b + h, k + 1) \), which comes from the fact that \( \log_b(b) = 1 \).
5. Sketch the curve through these points, bending it toward the asymptote on one side and flattening out gradually on the other, matching whether the function increases or decreases.

Start with the domain: \( x - 2 > 0 \), so \( x > 2 \). The vertical asymptote is the line \( x = 2 \). The parent point \( (1, 0) \) shifts right 2 and up 1, landing at \( (3, 1) \). The point \( (2, 1) \) on the parent graph shifts to \( (4, 2) \). Plotting these and following the same rising shape as \( y = \log_2(x) \) gives the transformed graph.

Notice that finding where this graph crosses the x-axis means setting \( y = 0 \) and solving \( \log_2(x - 2) + 1 = 0 \) for \( x \), which is the same kind of work covered in solving logarithmic equations.

Starting from the parent graph \( y = \log_b(x) \), the equation \( y = a\log_b(x - h) + k \) produces these changes:

• \( h \) shifts the graph, and its asymptote, left or right.
• \( k \) shifts the graph up or down (the asymptote does not move vertically).
• \( a \) stretches or compresses the graph vertically; if \( a \) is negative, the graph reflects across the x-axis.
• A negative sign inside the log, as in \( \log_b(-x) \), reflects the graph across the y-axis.

• Forgetting that the domain never includes zero or negative numbers, since a logarithm of a non-positive number is undefined.
• Drawing the curve as if it touches or crosses the vertical asymptote; it must always approach it without touching.
• Mixing up horizontal and vertical shifts, since the number added or subtracted inside the log moves the graph horizontally, while a number added outside moves it vertically.
• Assuming every log graph increases; when the base is between 0 and 1, the graph decreases instead.

Once you're confident reading a graph's shape and key points, it's worth practicing the reverse skill too: finding a logarithmic function given its graph asks you to work backward from a picture to an equation.

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