Properties of Logarithms

Key Terms

Domain

Inverse Relationship with Exponential Functions

Summary of Properties

Properties of Logarithms FAQ

Structural Characteristics of Logarithmic Functions

Every logarithmic function

f(x) = \log_a(x)

possesses properties that govern its behavior across the domain. These properties — domain, range, monotonicity, injectivity, continuity, asymptotic behavior, and inverse relationship to exponentials — determine how logarithms interact with equations, inequalities, and graphs.

Key Terms

Structural Properties

Argument (of a Logarithm)— must be positive; determines the domain

Monotonicity— increasing for base

> 1

, decreasing for base between

0

and

1

One-to-One Property— distinct inputs always produce distinct outputs

Logarithmic Function— domain, range, asymptote, and inverse relationship

See All Algebra Definitions →

Domain

The domain of

f(x) = \log_a(x)

is the set of all positive real numbers:

(0, \infty)

.

This restriction follows from the definition. The equation

\log_a(x) = y

means

a^y = x

. Since

a > 0

and

a^y > 0

for all real

y

, the output of any exponential with positive base is strictly positive. No real exponent produces zero or a negative number.

Consequently,

\log_a(0)

is undefined — no solution to

a^y = 0

exists. Similarly,

\log_a(-5)

is undefined — no real

y

satisfies

a^y = -5

.

For composite arguments, the entire expression inside the logarithm must be positive. The function

\log_2(x - 3)

requires

x - 3 > 0

, giving domain

x > 3

. The function

\log_5(x^2 + 1)

has domain all real numbers, since

x^2 + 1 > 0

for every

x

The argument sets the domain

The section's two composite logarithms. log₂(x − 3) needs x − 3 > 0, so its graph begins at the dashed line x = 3 and nothing exists to the left. log₅(x² + 1) has an argument that is positive for every real x, so its graph runs across the whole axis, touching its lowest value 0 at x = 0.

The domain of a composite logarithm is wherever its whole argument is positive.

Range

The range of

f(x) = \log_a(x)

is all real numbers:

(-\infty, \infty)

.

As

x

increases from just above zero toward infinity,

\log_a(x)

takes every real value exactly once. Near zero, the logarithm plunges toward

-\infty

. At

x = 1

, the logarithm equals zero. As

x

grows large, the logarithm increases without bound, though slowly.

This unlimited range contrasts with the exponential function

g(x) = a^x

, whose range is

(0, \infty)

. The domain and range swap between a function and its inverse — logarithms and exponentials exhibit this exchange precisely.

Monotonicity

Logarithmic functions are strictly monotonic — either always increasing or always decreasing throughout their domain.

When

a > 1

, the function

f(x) = \log_a(x)

is strictly increasing. If

x_1 < x_2

, then

\log_a(x_1) < \log_a(x_2)

. Larger inputs produce larger outputs. The functions

\log_2(x)

\log_{10}(x)

, and

\ln(x)

all increase.

When

0 < a < 1

, the function

f(x) = \log_a(x)

is strictly decreasing. If

x_1 < x_2

, then

\log_a(x_1) > \log_a(x_2)

. Larger inputs produce smaller outputs. The function

\log_{1/2}(x)

decreases.

This property is critical for solving inequalities. Taking logarithms preserves inequality direction when

a > 1

and reverses it when

0 < a < 1

One-to-One Property

Logarithmic functions are one-to-one (injective): distinct inputs always produce distinct outputs.

Algebraically: if

\log_a(x) = \log_a(y)

, then

x = y

.

This follows directly from strict monotonicity. A function that is always increasing (or always decreasing) cannot assign the same output to two different inputs — it would have to "turn around," violating monotonicity.

The one-to-one property justifies a key technique for solving equations. When an equation has the form

\log_a(M) = \log_a(N)

with the same base on both sides, the arguments must be equal:

M = N

. This reduces a logarithmic equation to an algebraic one.

Example:

\log_3(2x + 1) = \log_3(x + 5)

implies

2x + 1 = x + 5

, giving

x = 4

.

The logarithmic equation solver uses this property whenever both sides are single logarithms with the same base.

One-to-one, seen with horizontal lines

Every horizontal line meets log₂ x exactly once, because the curve only ever rises: no two inputs share an output, so log₂(M) = log₂(N) forces M = N. The function log₂(x² + 1) is a different story: its argument falls and then rises again, so the graph turns back at 0 and the line y = 1 meets it at x = ±1. Two inputs share an output, and the one-to-one property fails.

This is why equating arguments is safe for log_a(x) but needs care when the argument is not monotonic.

Continuity

The function

f(x) = \log_a(x)

is continuous on its entire domain

(0, \infty)

.

There are no jumps, breaks, or holes in the graph for any

x > 0

. The function transitions smoothly from one value to the next. Small changes in input produce small changes in output.

At the boundary

x = 0

, continuity fails — the function is not defined there. The graph approaches the

y

-axis but never reaches it. This boundary behavior leads to the vertical asymptote discussed below.

Asymptotic Behavior

The line

x = 0

(the

y

-axis) is a vertical asymptote for every logarithmic function.

As

x \to 0^+

(approaching zero from the right):

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

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

The graph approaches the

y

-axis without touching it, plunging downward (for

a > 1

) or rising upward (for

0 < a < 1

).

As

x \to \infty

\lim_{x \to \infty} \log_a(x) = +\infty \quad \text{when } a > 1

\lim_{x \to \infty} \log_a(x) = -\infty \quad \text{when } 0 < a < 1

Growth toward infinity is slow. For

a > 1

, the logarithm increases without bound but at a decelerating rate — doubling the input adds only a fixed constant to the output.

Inverse Relationship with Exponential Functions

The functions

f(x) = \log_a(x)

and

g(x) = a^x

are inverses of each other.

Composition in either order returns the input:

f(g(x)) = \log_a(a^x) = x \quad \text{for all real } x

g(f(x)) = a^{\log_a(x)} = x \quad \text{for all } x > 0

Graphically, the curves

y = \log_a(x)

and

y = a^x

are reflections of each other across the line

y = x

. Points

(p, q)

on one curve correspond to points

(q, p)

on the other.

Domain and range exchange:

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

Asymptotes exchange orientation:

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

This inverse relationship provides the foundation for converting between logarithmic and exponential forms when solving equations.

Feature	logₐ(x)	aˣ
Domain	(0, ∞)	(−∞, ∞)
Range	(−∞, ∞)	(0, ∞)
Asymptote	vertical, x = 0	horizontal, y = 0
Anchor point	(1, 0)	(0, 1)
Composition with the other	logₐ(aˣ) = x	a^(logₐ(x)) = x

Summary of Properties

The seven properties above together describe how logarithmic functions behave on their domain and how that behavior shapes work with logarithmic equations, inequalities, and graphs. The table below collects each property in a single row — the statement and the practical consequence — useful as a reference card.

Property	Statement	Practical consequence
Domain	(0, ∞) — the argument must be strictly positive	verify the argument is positive before accepting any solution
Range	(−∞, ∞) — every real value is attained	logarithms can equal any real number, positive or negative
Monotonicity	strictly increasing for a > 1; strictly decreasing for 0 < a < 1	inequality direction is set by the base
One-to-one	logₐ(M) = logₐ(N) implies M = N	drop matching logs to reduce a logarithmic equation to an algebraic one
Continuity	continuous on (0, ∞) — no jumps, breaks, or holes	small changes in input produce small changes in output
Vertical asymptote	x = 0; logₐ(x) → ∓∞ as x → 0⁺ (sign depends on base)	graph hugs the y-axis but never touches it
Inverse of aˣ	logₐ(aˣ) = x and a^(logₐ(x)) = x	switch between exponential and logarithmic form to solve equations

Properties of Logarithms FAQ

Why is the logarithm of a negative number undefined?

Because

\log_a(x) = y

means

a^y = x

, and a positive base raised to any real power is positive. No exponent

y

satisfies

2^y = -5

, so

\log_2(-5)

has no answer. The same reasoning rules out zero, since nothing satisfies

a^y = 0

. Negative and zero inputs are excluded by the definition, not by convention.Read more →

How do you find the domain of something like $\log_2(x - 3)$ ?

Set whatever sits inside the logarithm greater than zero and solve. Here

x - 3 > 0

gives domain

x > 3

. The rule applies to the whole argument, not just to

x

, so

\log_5(x^2 + 1)

has domain all real numbers because

x^2 + 1

is positive for every

x

. Strictly greater than zero, never equal to it.Read more →

Why does a logarithm grow so slowly?

Because each fixed step in the output demands a multiplicative jump in the input. With

\log_2

, moving the output from

10

11

requires doubling the input, and doubling again adds only one more. Put the other way round, multiplying the input by a constant adds a constant to the output. The curve rises without bound, but at an ever-decelerating rate.Read more →

Why does $\log_a(a^x) = x$ hold for every $x$ , but $a^{\log_a(x)} = x$ only for $x > 0$ ?

Because the two compositions begin in different places. In

\log_a(a^x)

the exponential runs first and accepts any real

x

, handing the logarithm a positive number, so nothing is excluded. In

a^{\log_a(x)}

the logarithm runs first and demands a positive input, so

x \leq 0

never gets in. The restriction belongs to whichever function goes first.Read more →

Properties of Logarithms

Structural Characteristics of Logarithmic Functions

Key Terms

Structural Properties

Domain

Range

Monotonicity

One-to-One Property

Continuity

Asymptotic Behavior

Inverse Relationship with Exponential Functions

Summary of Properties

Properties of Logarithms FAQ

Why is the logarithm of a negative number undefined?

How do you find the domain of something like log⁡2(x−3)\log_2(x - 3)log2​(x−3)?

Why does a logarithm grow so slowly?

Why does log⁡a(ax)=x\log_a(a^x) = xloga​(ax)=x hold for every xxx, but alog⁡a(x)=xa^{\log_a(x)} = xaloga​(x)=x only for x>0x > 0x>0?

How do you find the domain of something like $\log_2(x - 3)$ ?

Why does $\log_a(a^x) = x$ hold for every $x$ , but $a^{\log_a(x)} = x$ only for $x > 0$ ?