Logarithmic graphs: plot and transform y = a ln x + c

A logarithmic graph is the graph of y = log x, or a stretch or shift of it. It only exists for x greater than 0, meets the x-axis at x = 1, climbs steeply near the y-axis and flattens as x grows, and gets closer and closer to the y-axis without touching it: the y-axis is its vertical asymptote. Our logarithmic graph calculator plots y = a ln(x) + c. Change a and c to stretch, flip and shift the curve, and it finds where the curve crosses the x-axis.

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Logarithmic Graph

Plot and analyse logarithmic functions

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How to use the logarithmic graph calculator

1. Open the Logarithmic Graph. It opens on y = ln(x) − 1, shown in the preview.
2. Type a, the number in front of ln(x), in the box marked A, and c, the number added on the end, in the box marked C. The curve redraws as you type.
3. Under Transform, stretch the curve (a × 2), compress it (a × 0.5) or reflect it in the x-axis (a × −1).
4. Read the Intercepts table for the x-intercept, and use Analysis for the derivative.
5. Press Random problem for questions on log graphs.

The log in this calculator is the natural logarithm, ln, base e ≈ 2.718; the preview writes it as ln. For a base 10 curve, type log10(x) into the Custom Graph.

The shape of a logarithmic graph

Here x is across and y = ln x is up. These features are true of every graph y = log_b x with b > 1:

• Domain: x > 0. There is no log of 0 or of a negative number, so there is no curve on the left.
• Vertical asymptote: x = 0. As x gets close to 0, y heads down without limit.
• x-intercept: (1, 0), because log 1 = 0 in any base.
• No y-intercept, since x = 0 isn't allowed.
• Always increasing, ever more slowly: the gradient of ln x is 1/x, which shrinks as x grows.
• Range: all real numbers. It reaches any height, just very slowly.

A log graph is the reflection of the exponential graph in the line y = x: y = ln x and y = eˣ are inverses, so swapping x and y turns one into the other. The exponential graphs guide covers the other half.

How a and c transform y = a ln x + c

None of these move the asymptote: it stays at x = 0. Setting y = 0 gives the x-intercept:

Worked examples you can check in the calculator

Example 1: the opening graph

y = ln x − 1 is the basic curve moved down 1. Its x-intercept is where ln x = 1, so x = e. The Intercepts table shows (2.718281828459, 0) and says there is no y-intercept. The derivative, 1/x, is shown under Analysis.

Example 2: a stretch and a shift

Type a = 2 and c = −4: y = 2 ln x − 4. The x-intercept is x = e^4/2 = e² = 7.389; the table gives 7.389056098931. The curve is twice as steep as ln x at every x, since its derivative is 2/x.

Example 3: a reflection

Press reflect on y = ln x, or type a = −1 and c = 0: y = −ln x. It falls from left to right, still crosses at (1, 0), and still hugs the y-axis, now from above.

Example 4: an intercept between 0 and 1

Type a = 3 and c = 1: y = 3 ln x + 1. The x-intercept is x = e^−1/3 = 0.7165; the table gives 0.716531310574. With c positive, the curve has been lifted, so it crosses the axis before x = 1.

Logarithmic scales

"Logarithmic graph" sometimes means a graph drawn on a logarithmic scale, where each step along an axis multiplies by the same amount, such as 1, 10, 100, 1,000. That is a different idea: an exponential relationship such as y = 3 × 10ˣ looks like a straight line when y is on a log scale. The calculator draws log functions on ordinary axes; it doesn't draw log-scale axes. To use the idea by hand, take logs: log y = log 3 + x, which is a straight line with gradient 1 when you plot log y against x.

Common mistakes

• Drawing the curve for x ≤ 0. It doesn't exist there.
• Letting it touch the y-axis. It only gets closer.
• Moving the asymptote with c. Adding c moves the graph up or down; the asymptote x = 0 stays put.
• Thinking it levels off. It keeps rising forever, just slowly.
• Mixing up ln and log. ln is base e. In this calculator, log means ln too; base 10 is log10.

Read next

Maths · GraphsExponential graphs: y = aˣ, eˣ, growth and decayHow exponential graphs work: the shape of y = aˣ, the asymptote, the point (0, 1), growth and decay, y = eˣ, transformations and log graphs.