, and extending infinitely in both vertical directions. The base determines whether the curve rises or falls, and transformations shift, stretch, and reflect the basic shape. Understanding these graphs reinforces the properties of logarithms and provides geometric insight into inequalities.

Key Terms

Graph Features

Logarithmic Function— domain

(0, \infty)

, range

(-\infty, \infty)

, passes through

(1, 0)

Base (of a Logarithm)— determines whether the curve rises or falls

Monotonicity—

a > 1

gives an increasing curve;

0 < a < 1

gives a decreasing curve

Anchor Values

Logarithm—

\log_a(1) = 0

and

\log_a(a) = 1

anchor every graph

See All Algebra Definitions →

The Basic Shape

The graph of

y = \log_a(x)

for

a > 1

rises slowly from left to right. Starting from extreme negative

y

-values near the

y

-axis, the curve crosses through

(1, 0)

, continues through

(a, 1)

, and extends upward without bound as

x

increases.

The curve is concave down throughout — it bends toward the

x

-axis. Growth slows as

x

increases: the gap between

\log_a(10)

and

\log_a(100)

equals the gap between

\log_a(1)

and

\log_a(10)

. Each multiplication of

x

a

adds exactly

1

to the output.

For

0 < a < 1

, the shape mirrors vertically. The curve descends from upper left toward lower right, still passing through

(1, 0)

but now through

(a, 1)

where

a < 1

means this point lies to the left of

x = 1

Equal steps from doubling

log₂ x passes through (1, 0), (2, 1), (4, 2) and (8, 3): every time x doubles, the output goes up by exactly 1. That is why the curve keeps rising forever yet flattens out: reaching the next unit of height takes twice as long a stretch of x as the last one.

Concave down everywhere: each step up costs more x than the one before.

Domain and Range on the Graph

The domain

(0, \infty)

appears as the horizontal extent of the graph. The curve exists only for

x > 0

— nothing appears on or to the left of the

y

-axis.

The range

(-\infty, \infty)

appears as the vertical extent. The curve spans all

y

-values, extending upward and downward without bound.

No horizontal asymptote exists. As

x \to \infty

, the function continues increasing (for

a > 1

) or decreasing (for

0 < a < 1

), never leveling off. This contrasts with exponential functions, which have horizontal asymptotes.

The Vertical Asymptote

The line

x = 0

is a vertical asymptote for every logarithmic function.

As

x \to 0^+

\lim_{x \to 0^+} \log_a(x) = -\infty \quad \text{for } a > 1

\lim_{x \to 0^+} \log_a(x) = +\infty \quad \text{for } 0 < a < 1

The graph approaches the

y

-axis but never touches or crosses it. For

a > 1

, the curve plunges downward as it nears the asymptote. For

0 < a < 1

, the curve rises upward.

Under horizontal transformations, the asymptote shifts. The function

y = \log_a(x - h)

has vertical asymptote at

x = h

Opposite directions at the same wall

Near x = 0 both curves run along the y-axis without touching it. log₂ x plunges toward −∞ and log₁⁄₂ x climbs toward +∞, but neither ever reaches x = 0, where no power of the base gives 0. Away from the axis the two are mirror images across the x-axis.

Only a horizontal shift can move this asymptote.

Key Points

Three points anchor any logarithmic graph:

(1, 0)

: Every logarithm satisfies

\log_a(1) = 0

. This point lies on every logarithmic curve regardless of base.

(a, 1)

: Every logarithm satisfies

\log_a(a) = 1

. For

y = \log_2(x)

, this is

(2, 1)

. For

y = \log_{10}(x)

, this is

(10, 1)

. For

y = \ln(x)

, this is

(e, 1) \approx (2.718, 1)

(1/a, -1)

: Since

\log_a(1/a) = \log_a(a^{-1}) = -1

. For

y = \log_2(x)

, this is

(0.5, -1)

.

When sketching, plot these points first. The curve passes through all three, approaching the vertical asymptote as

x \to 0^+

Anchor point	Why it lies on every graph	Concrete coordinates
(1, 0)	logₐ(1) = 0 because a⁰ = 1 for every valid base	(1, 0) for log₂, ln, log₁₀ — the same point on all of them
(a, 1)	logₐ(a) = 1 because a¹ = a	(2, 1) for log₂; (e, 1) ≈ (2.718, 1) for ln; (10, 1) for log₁₀
(1⁄a, −1)	logₐ(1⁄a) = −1 because a⁻¹ = 1⁄a	(0.5, −1) for log₂; (1⁄e, −1) ≈ (0.368, −1) for ln; (0.1, −1) for log₁₀

Effect of Base on Shape

The base controls both direction and steepness.

For

a > 1

: the function increases. Larger bases produce flatter curves. The graph of

y = \log_{10}(x)

rises more slowly than

y = \log_2(x)

because base

10

requires larger

x

-values to achieve the same output.

For

0 < a < 1

: the function decreases. The graph of

y = \log_{1/2}(x)

is the reflection of

y = \log_2(x)

across the

x

-axis. In general,

\log_{1/a}(x) = -\log_a(x)

.

Comparing common and natural logarithms:

\ln(x)

rises faster than

\log(x)

because

e < 10

. The natural logarithm reaches output

1

x = e \approx 2.718

, while the common logarithm reaches

1

x = 10

.

The table below pins down how four specific bases differ when graphed on the same axes — direction, anchor location, and where each falls in the steepness ordering.

Function	Direction	Anchor (a, 1)	Steepness vs the others
y = log₂(x)	increasing	(2, 1)	smallest base of the three increasing logs — steepest
y = ln(x)	increasing	(e, 1) ≈ (2.72, 1)	between log₂ and log₁₀ — middle of the pack
y = log₁₀(x)	increasing	(10, 1)	largest base of the three — flattest
y = log₍₁⁄₂₎(x)	decreasing	(0.5, 1)	reflection of log₂ across the x-axis (since log₍₁⁄₂₎(x) = −log₂(x))

Horizontal Shifts

The function

y = \log_a(x - h)

shifts the graph horizontally by

h

units.

When

h > 0

, the shift is rightward. The vertical asymptote moves from

x = 0

x = h

. The point

(1, 0)

moves to

(1 + h, 0)

.

When

h < 0

, the shift is leftward. The asymptote moves to

x = h

(a negative value), and the domain becomes

x > h

.

Example:

y = \log_2(x - 3)

has asymptote at

x = 3

, passes through

(4, 0)

and

(5, 1)

, and has domain

x > 3

.

Example:

y = \ln(x + 2)

has asymptote at

x = -2

, passes through

(-1, 0)

and

(e - 2, 1)

, and has domain

x > -2

Moving the asymptote

The section's log₂(x − 3) against the original log₂ x (dashed). Every point slides 3 units right: the asymptote moves from x = 0 to x = 3, (1, 0) becomes (4, 0) and (2, 1) becomes (5, 1). The domain becomes x > 3.

Subtracting inside the logarithm moves the graph right, not left.

Vertical Shifts

The function

y = \log_a(x) + k

shifts the graph vertically by

k

units.

When

k > 0

, the shift is upward. When

k < 0

, the shift is downward.

The vertical asymptote remains at

x = 0

— vertical shifts do not affect the asymptote location. The domain remains

x > 0

.

Key points shift accordingly:

(1, 0)

becomes

(1, k)

, and

(a, 1)

becomes

(a, 1 + k)

.

Example:

y = \log_3(x) + 2

passes through

(1, 2)

and

(3, 3)

, with asymptote still at

x = 0

.

Example:

y = \ln(x) - 1

passes through

(1, -1)

and

(e, 0)

Lifting the curve

The section's log₃ x + 2 against log₃ x. Every point rises 2 units: (1, 0) becomes (1, 2) and (3, 1) becomes (3, 3). The asymptote is a vertical line, so a vertical shift leaves it at x = 0, and the domain is still x > 0.

Vertical changes act on outputs only; the domain is untouched.

Stretches and Compressions

The function

y = c \cdot \log_a(x)

stretches or compresses the graph vertically.

When

|c| > 1

, the graph stretches away from the

x

-axis. When

0 < |c| < 1

, the graph compresses toward the

x

-axis.

The point

(1, 0)

remains fixed — multiplying zero by any constant gives zero. The point

(a, 1)

moves to

(a, c)

.

Example:

y = 2\log_2(x)

passes through

(1, 0)

and

(2, 2)

. The graph is stretched vertically by factor

2

.

Example:

y = \frac{1}{2}\ln(x)

passes through

(1, 0)

and

(e, 0.5)

. The graph is compressed vertically by factor

1/2

.

The asymptote remains at

x = 0

, and the domain remains

x > 0

A vertical stretch by 2

The section's 2 log₂ x against log₂ x. Every output is doubled, so (2, 1) becomes (2, 2) and the curve is steeper everywhere. The point (1, 0) cannot move, because doubling 0 still gives 0, and the asymptote stays at x = 0.

A negative factor would add a reflection across the x-axis.

Reflections

Reflections flip the graph across an axis.

y = -\log_a(x)

: reflection across the

x

-axis. The point

(a, 1)

becomes

(a, -1)

. An increasing logarithm becomes decreasing, and vice versa. This is equivalent to changing the base:

-\log_a(x) = \log_{1/a}(x)

y = \log_a(-x)

: reflection across the

y

-axis. The domain changes from

x > 0

x < 0

. The asymptote moves from

x = 0

x = 0

(still the

y

-axis, approached from the left). The point

(a, 1)

becomes

(-a, 1)

.

Example:

y = -\log_2(x)

passes through

(1, 0)

(2, -1)

, and

(4, -2)

.

Example:

y = \log_2(-x)

passes through

(-1, 0)

(-2, 1)

, and has domain

x < 0

Two different reflections

The section's −log₂ x and log₂(−x) against log₂ x (dashed). Negating the output flips the curve across the x-axis: (2, 1) becomes (2, −1) and the function now decreases. Negating the input flips it across the y-axis: (2, 1) becomes (−2, 1) and the domain becomes x < 0.

The sign outside affects heights; the sign inside affects positions along x.

Writing Equations from Graphs

To determine the equation of a transformed logarithmic graph:

Step 1: Identify the vertical asymptote. If at

x = h

, the function includes

(x - h)

as the argument.

Step 2: Find a point on the graph, typically where the curve crosses a convenient location. Use this to determine the base or verify it.

Step 3: Check for vertical stretch/compression by comparing the vertical distance between known points.

Step 4: Check for vertical shift by noting if

(1 + h, k)

lies on the curve for

k \neq 0

.

Example: A graph has asymptote at

x = 2

, passes through

(3, 0)

and

(4, 1)

.

Asymptote at

x = 2

suggests

y = \log_a(x - 2)

. The point

(3, 0)

confirms:

\log_a(3 - 2) = \log_a(1) = 0

. The point

(4, 1)

gives:

\log_a(2) = 1

, so

a = 2

.

Equation:

y = \log_2(x - 2)

Inverse Relationship with Exponential Graphs

The graphs of

y = \log_a(x)

and

y = a^x

are reflections of each other across the line

y = x

Every point

(p, q)

on the logarithmic curve corresponds to a point

(q, p)

on the exponential curve. The curves intersect the line

y = x

at the same location only if

a^c = c

for some

c

Asymptote correspondence:

The exponential $y = a^x$ has horizontal asymptote $y = 0$
The logarithm $y = \log_a(x)$ has vertical asymptote $x = 0$

Domain and range exchange:

$a^x$ : domain $(-\infty, \infty)$ , range $(0, \infty)$
$\log_a(x)$ : domain $(0, \infty)$ , range $(-\infty, \infty)$

Graphing both functions on the same axes with the line

y = x

demonstrates this inverse relationship visually. The symmetry confirms that each function undoes the other.

Summary of Transformations

The transformations covered above — horizontal and vertical shifts, vertical stretches, and reflections across each axis — combine in any sequence to produce arbitrary logarithmic graphs. The table below collects each transformation in one place: its form, its effect on the graph, and the new location of the anchor point

(a, 1)

. Use it as a reference while sketching transformed graphs or reading equations off of them.

Transformation	Form	Effect on the graph	(a, 1) becomes
Horizontal shift	y = logₐ(x − h)	shifts h units right (h > 0); asymptote moves to x = h; domain becomes x > h	(a + h, 1)
Vertical shift	y = logₐ(x) + k	shifts k units up (k > 0); asymptote and domain unchanged	(a, 1 + k)
Vertical stretch	y = c · logₐ(x)	scales y-values by factor c; (1, 0) stays fixed; \|c\| > 1 stretches, \|c\| < 1 compresses	(a, c)
Reflection in x-axis	y = −logₐ(x)	flips up ↔ down; an increasing log becomes decreasing; equivalent to log_{1/a}	(a, −1)
Reflection in y-axis	y = logₐ(−x)	flips left ↔ right; domain becomes x < 0; asymptote still the y-axis	(−a, 1)

Logarithmic Graphs FAQ

Is $-\log_2(x)$ the same as $\log_{1/2}(x)$ ?

Yes, they are the same function. Reflecting a logarithm across the

x

-axis is identical to inverting its base:

-\log_a(x) = \log_{1/a}(x)

. Both pass through

(1, 0)

and send

(2, 1)

(2, -1)

. This is why every base between

0

and

1

produces a decreasing curve, since such a graph is the mirror image of the increasing one with the reciprocal base.Read more →

Is stretching a logarithmic graph the same as changing its base?

Yes. A vertical stretch by a factor

c

is exactly a change of base, since

c\log_a(x) = \log_{a^{1/c}}(x)

, so

2\log_2(x)

is the same function as

\log_{\sqrt{2}}(x)

. Steepness has a single source: every logarithm is a constant multiple of

\ln(x)

. Different bases are not different shapes, only one curve at different vertical scales.Read more →

Why does $\log_2(-x)$ have a graph at all?

Because

-x

is positive whenever

x

is negative, and it is the argument that must be positive, not the variable. The domain becomes

x < 0

, so the curve lies entirely left of the

y

-axis, passing through

(-1, 0)

and

(-2, 1)

. The asymptote is still the

y

-axis, now approached from the left.Read more →

Logarithmic Graphs

Visualizing the Inverse of Exponentials

Key Terms

Graph Features

Anchor Values

The Basic Shape

Domain and Range on the Graph

The Vertical Asymptote

Key Points

Effect of Base on Shape

Horizontal Shifts

Vertical Shifts

Stretches and Compressions

Reflections

Writing Equations from Graphs

Inverse Relationship with Exponential Graphs

Summary of Transformations

Logarithmic Graphs FAQ

Is −log⁡2(x)-\log_2(x)−log2​(x) the same as log⁡1/2(x)\log_{1/2}(x)log1/2​(x)?

Is stretching a logarithmic graph the same as changing its base?

Why does log⁡2(−x)\log_2(-x)log2​(−x) have a graph at all?

Is $-\log_2(x)$ the same as $\log_{1/2}(x)$ ?

Why does $\log_2(-x)$ have a graph at all?