Logarithmic Function Grapher

About the Logarithmic Function Grapher

Plot logarithmic functions like y = log(x) or y = ln(x) and see their domain and asymptote.

How to use this calculator

1. Enter a logarithmic function such as y = log(x), y = ln(x), or y = log_2(x).
2. If your function has shifts or stretches, include them, for example y = ln(x - 3) + 2.
3. Look at the graph to identify the vertical asymptote and the x-values where the curve is defined.
4. Use the plotted curve and displayed results to compare different logarithmic functions and their transformations.

The formula explained

A logarithmic function finds the exponent needed to produce a given number, for example ln(x) uses base e and log(x) usually means base 10 unless stated otherwise. The grapher plots that relationship so you can see its shape, domain, and asymptote.

• x = the input value, which must be positive for a basic logarithmic function
• y = the output value of the logarithmic function
• b = the base of the logarithm, such as 10, e, or 2
• h = a horizontal shift inside the logarithm, such as in log(x - h)

Step by step method

1. Choose the logarithmic expression you want to graph, for example y = ln(x - 3) + 1.
2. Check the inside of the logarithm first, because it must be greater than 0, so x - 3 > 0.
3. Find the vertical asymptote by setting the inside equal to 0, so x - 3 = 0 gives x = 3.
4. Use the graph to confirm the curve approaches the asymptote and rises or falls as expected.

Worked example

Problem. Graph y = ln(x - 3) + 1 and identify its domain and vertical asymptote.

1. Start with the inside of the log, x - 3, and require x - 3 > 0.
2. Solve the inequality, x > 3, so the domain is all x-values greater than 3.
3. Set x - 3 = 0 to find the vertical asymptote, which gives x = 3. The graph is shifted right 3 units and up 1 unit from y = ln(x).

Tips and common mistakes

• Do not forget that the logarithm input must be positive, not zero or negative.
• If the graph looks shifted left or right, check the sign inside the parentheses carefully, because y = ln(x + 2) shifts left 2, while y = ln(x - 2) shifts right 2.

Frequently asked questions