Graphs of logarithmic functions

Logarithm graphs are exponential graphs reflected in y = x: y = ln x crosses the x-axis at x = 1, has the y-axis as an asymptote and grows ever more slowly. These six cover ln x, other bases and common transformations.

Every value in the key-features tables is checked by script against the Grapher's own calculations (roots, turning points, asymptotes, domain and range).

Graph of y = ln x

The natural logarithm, the inverse of y = eˣ: defined only for x > 0, crossing the x-axis at 1 and falling without limit as x → 0.

Open y = ln x in the Grapher

Key features

Domain	x > 0
Range	all real numbers (y ∈ ℝ)
x-intercept	(1, 0)
y-intercept	none (x = 0 is not in the domain)
Turning points	none
Asymptotes	vertical: x = 0 (the y-axis)
Symmetry	neither even nor odd

How to sketch it

1. Draw the asymptote first as dashed lines: x = 0 (the y-axis).
2. Mark the x-intercept (1, 0).
3. Only draw the curve where it is defined: x > 0.
4. Check the range: all real numbers (y ∈ ℝ). Your sketch should not go anywhere else.

y = ln x is the reflection of y = eˣ in the line y = x (its inverse): see the graph of y = eˣ. Graphs made from this one: y = log₁₀ x, y = ln(x + 2), y = ln|x|. Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.

Graph of y = log₁₀ x

The base-10 logarithm: log x = 1 at x = 10 and 2 at x = 100. Same shape as ln x, squashed vertically.

Open y = log₁₀ x in the Grapher

Key features

Domain	x > 0
Range	all real numbers (y ∈ ℝ)
x-intercept	(1, 0)
y-intercept	none (x = 0 is not in the domain)
Turning points	none
Asymptotes	vertical: x = 0 (the y-axis)
Symmetry	neither even nor odd

How to sketch it

1. Draw the asymptote first as dashed lines: x = 0 (the y-axis).
2. Mark the x-intercept (1, 0).
3. Only draw the curve where it is defined: x > 0.
4. Check the range: all real numbers (y ∈ ℝ). Your sketch should not go anywhere else.

y = log₁₀ x is log x = ln x / ln 10: y = ln x stretched vertically by scale factor 1/ln 10 ≈ 0.434: see the graph of y = ln x. Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.

Graph of y = log₂ x

The base-2 logarithm, the inverse of y = 2ˣ: log₂ x = 3 at x = 8.

Open y = log₂ x in the Grapher

Key features

Domain	x > 0
Range	all real numbers (y ∈ ℝ)
x-intercept	(1, 0)
y-intercept	none (x = 0 is not in the domain)
Turning points	none
Asymptotes	vertical: x = 0 (the y-axis)
Symmetry	neither even nor odd

How to sketch it

1. Draw the asymptote first as dashed lines: x = 0 (the y-axis).
2. Mark the x-intercept (1, 0).
3. Only draw the curve where it is defined: x > 0.
4. Check the range: all real numbers (y ∈ ℝ). Your sketch should not go anywhere else.

y = log₂ x is the reflection of y = 2ˣ in the line y = x (its inverse): see the graph of y = 2ˣ. Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.

Graph of y = ln(x + 2)

The graph of y = ln x moved 2 to the left, so the asymptote moves to x = −2.

Open y = ln(x + 2) in the Grapher

Key features

Domain	x > −2
Range	all real numbers (y ∈ ℝ)
x-intercept	(−1, 0)
y-intercept	(0, ln 2) ≈ (0, 0.693)
Turning points	none
Asymptotes	vertical: x = −2
Symmetry	neither even nor odd

How to sketch it

1. Draw the asymptote first as dashed lines: x = −2.
2. Mark the x-intercept (−1, 0), the y-intercept (0, ln 2) ≈ (0, 0.693).
3. Only draw the curve where it is defined: x > −2.
4. Check the range: all real numbers (y ∈ ℝ). Your sketch should not go anywhere else.

y = ln(x + 2) is y = ln x translated 2 units to the left (replace x with x + 2): see the graph of y = ln x. Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.

Graph of y = ln|x|

The graph of y = ln x together with its reflection in the y-axis: defined for every x except 0.

Open y = ln|x| in the Grapher

Key features

Domain	x ≠ 0
Range	all real numbers (y ∈ ℝ)
x-intercepts	(−1, 0) and (1, 0)
y-intercept	none (x = 0 is not in the domain)
Turning points	none
Asymptotes	vertical: x = 0 (the y-axis)
Symmetry	even function: f(−x) = f(x), symmetric about the y-axis

How to sketch it

1. Draw the asymptote first as dashed lines: x = 0 (the y-axis).
2. Mark the x-intercepts (−1, 0) and (1, 0).
3. Only draw the curve where it is defined: x ≠ 0.
4. Check the range: all real numbers (y ∈ ℝ). Your sketch should not go anywhere else.
5. Use the symmetry: draw the right-hand half and reflect it in the y-axis.

y = ln|x| is y = ln x for x > 0, reflected in the y-axis for x < 0: see the graph of y = ln x. Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.

Graph of y = x ln x

Defined for x > 0. The product rule gives dy/dx = ln x + 1, which is 0 at x = 1/e, the minimum. Near 0 the curve approaches the origin (it has no asymptote there).

Open y = x ln x in the Grapher

Key features

Domain	x > 0
Range	y ≥ −1/e (≈ −0.368)
x-intercept	(1, 0)
y-intercept	none (x = 0 is not in the domain)
Turning point	minimum (1/e, −1/e) ≈ (0.368, −0.368)
Asymptotes	none
Symmetry	neither even nor odd

How to sketch it

1. Mark the x-intercept (1, 0), the turning point (1/e, −1/e) ≈ (0.368, −0.368).
2. Only draw the curve where it is defined: x > 0.
3. Check the range: y ≥ −1/e (≈ −0.368). Your sketch should not go anywhere else.

Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.

The Grapher is our own free tool (graph.mathrevision.co.uk), built for IB, A Level and IGCSE students. It is not an exam calculator: practise on your own GDC for calculator papers.