Graph of ln x (natural logarithm)

Updated 20 October 2026 · Values computed by the ihateDesmos engine

The natural logarithm graph rises slowly forever but only exists to the right of the y-axis. It crosses the x-axis at x = 1, reaches 1 at x = e ≈ 2.718, and plunges toward −∞ as x approaches 0.

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The graph of

y=ln(x)

for −1 ≤ x ≤ 9. Dots mark its root at x = 1.

Key features

Domain: x > 0
Range: All real numbers
x-intercept: (1, 0)
Vertical asymptote: x = 0 (the y-axis): ln x → −∞ as x → 0⁺
Notable point: (e, 1), since ln e = 1
Monotonic: Increasing on its whole domain
Concavity: Concave down everywhere (ln″x = −1/x² < 0)
y-intercept: None: the graph never reaches x = 0
Derivative: 1x
Antiderivative: xln(x)−x+C

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

What the graph of ln x looks like

ln x is the inverse of eˣ, so its graph is the exponential curve reflected in the line y = x. Everything about eˣ flips: its horizontal asymptote y = 0 becomes the vertical asymptote x = 0, and its y-intercept (0, 1) becomes the x-intercept (1, 0).

The slope of ln x is 1/x. Near zero the graph is almost vertical; at x = 1 its slope is exactly 1; by x = 100 it is a mere 0.01. The curve keeps rising without bound, but extraordinarily slowly: ln(1,000,000) is only about 13.8.

The logarithm turns multiplication into addition, ln(ab) = ln a + ln b. On the graph, doubling x always adds the same amount, ln 2 ≈ 0.693, to the height, which is why equally spaced powers of 2 sit at equally spaced heights.

Points of interest

Found numerically to about twelve digits in the window −1 ≤ x ≤ 9, 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 ln x
x	y = ln x
1/10	−2.30259
1/2	−ln 2
1	0
2	ln 2
e	1
5	1.60944
10	2.30259

How to graph ln x by hand

1
Draw a dashed vertical asymptote along the y-axis.
2
Plot (1, 0), (e, 1) with e ≈ 2.72, and (e², 2) with e² ≈ 7.39.
3
Plot (1/e, −1) ≈ (0.37, −1) to see how fast the curve falls near the asymptote.
4
Draw a smooth increasing curve through the points that bends downward (concave down), hugging the y-axis as it falls.

Transformations

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

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y = ln(x - 2) Shifted right 2: the asymptote moves to x = 2 and the root to x = 3.
y = 2ln(x) Vertically stretched. This equals ln(x²) for x > 0.
y = ln(x) + 1 Shifted up 1: the root moves to x = 1/e ≈ 0.368.

Common mistakes

Drawing the graph to the left of the y-axis. ln x is undefined for x ≤ 0 (for real numbers).
Letting the curve level off to a horizontal asymptote. It has none; it grows without bound.
Confusing ln x (base e) with log x. In this calculator, as on most handheld calculators, log means base 10.

Where it shows up

Continuous growth and decay: the time for a quantity growing at rate r to multiply by k is ln(k)/r.
Logarithmic scales such as decibels and pH turn huge ranges into manageable numbers.
Information theory and statistics, where entropy and log-likelihoods are sums of logarithms.