Graph of log x (base 10 logarithm)

Updated 20 October 2026 · Values computed by the ihateDesmos engine

The graph of log x, the logarithm to base 10, has the same shape as every logarithm graph: it exists only for x > 0, crosses the x-axis at x = 1, and rises one unit each time x is multiplied by 10.

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The graph of

y=log(x)

for −2 ≤ x ≤ 22. Dots mark its root at x = 1.

Key features

Domain: x > 0
Range: All real numbers
x-intercept: (1, 0)
Vertical asymptote: x = 0
Notable points: (0.1, −1), (10, 1), (100, 2)
Monotonic: Increasing
Change of base: log x = ln x / ln 10 ≈ 0.4343 ln x
y-intercept: None: the graph never reaches x = 0
Derivative: 1xln(10)
Antiderivative: xln(x)−xln(10)+C

Derivative and antiderivative from the calculator’s algebra system. Domain, range and asymptotes are stated for real x.

What the graph of log x looks like

log x answers the question “10 to what power gives x?” So log 10 = 1, log 100 = 2, log 1000 = 3, and log 0.1 = −1. The graph rises by exactly one unit every time x is multiplied by 10, which makes it look almost flat after the first few units.

Every logarithm, whatever its base b > 1, has this shape. They are all vertical stretches of one another, since log_b x = ln x / ln b: a bigger base gives a flatter curve, and all of them pass through (1, 0) with an asymptote at x = 0.

For a base between 0 and 1, such as log base ½, the graph is reflected in the x-axis and decreases instead.

Points of interest

Found numerically to about twelve digits in the window −2 ≤ x ≤ 22, then shown in exact form where one exists.

x-intercepts (roots): x = 1
Local maxima: None in this window
Local minima: None in this window
Inflection points: None in this window

Table of values

A few points to plot, with exact values where they are simple.

Values of log x
x	y = log x
0.01	−2
1/10	−1
1	0
2	0.30103
5	0.69897
10	1
100	2

How to graph log x by hand

1
Draw the vertical asymptote x = 0.
2
Plot (1, 0) and (10, 1). If your axes allow it, plot (0.1, −1) too.
3
Add (2, 0.301) and (5, 0.699) to fix the curvature between 1 and 10.
4
Draw a smooth increasing curve, steep near the y-axis and very gentle beyond x = 10.

Transformations

How the graph moves when you change the formula. The original is log x in blue.

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y = ln(x)/ln(2) log base 2: steeper, reaching 1 at x = 2. Type log_2(x) in the calculator for the same graph.
y = log(x) + 1 Shifted up 1. This equals log(10x).
y = -log(x) Reflected in the x-axis: the same graph as log base 1/10.

Where it shows up

Decibels: sound level in dB is 10 log(I/I₀).
pH = −log[H⁺] in chemistry, and the Richter magnitude scale for earthquakes.
Log scales on plots, which make exponential growth look like a straight line.

Frequently asked questions

What base does log x use?

In this calculator, as on most handheld calculators, log x means base 10. Use ln x for base e, or log_b(x) for any base b.

What does the graph of a logarithmic function look like?

An increasing curve (for a base above 1) that exists only for x > 0, crosses the x-axis at (1, 0), has a vertical asymptote at x = 0 and rises more and more slowly.

What is log 0?

Undefined. As x approaches 0 from the right, log x goes to −∞.

How do I graph a log with a different base?

Use the change-of-base rule log_b x = log x / log b (or ln x / ln b). Every log graph is a vertical stretch of every other.