What is the domain of a logarithmic function?

The domain is (0, ∞) — all positive real numbers. Logarithms are undefined for zero and negative numbers because no real exponent makes a positive base equal zero or a negative number.

What is the range of a logarithmic function?

The range is (-∞, ∞) — all real numbers. As x approaches 0 from the right, log(x) goes to negative infinity. As x grows large, log(x) increases without bound.

Why is log of a negative number undefined?

Because a^y > 0 for any positive base a and any real exponent y. No real power of a positive number can produce a negative result, so log_a(-5) has no solution.

When is a logarithmic function increasing or decreasing?

For base > 1 (like log₂, log₁₀, ln), the function is strictly increasing. For base between 0 and 1 (like log_{1/2}), the function is strictly decreasing.

What is the one-to-one property of logarithms?

If log_a(x) = log_a(y), then x = y. Each output corresponds to exactly one input. This property lets you set arguments equal when logs with the same base are equal.

What is the vertical asymptote of log(x)?

The line x = 0 (y-axis) is a vertical asymptote. As x approaches 0 from the right, log(x) approaches negative infinity (for base > 1). The graph never touches x = 0.

How are logarithmic and exponential functions related?

They are inverses. log_a(a^x) = x for all real x, and a^(log_a(x)) = x for x > 0. Their graphs are reflections across y = x, and their domains/ranges are swapped.

Is log(x) continuous?

Yes, log_a(x) is continuous on its entire domain (0, ∞). There are no jumps, breaks, or holes. Small changes in x produce small changes in log(x).

Why does monotonicity matter for solving inequalities?

Monotonicity determines if inequality direction is preserved or reversed. For base > 1 (increasing), direction is preserved. For base between 0 and 1 (decreasing), direction reverses.

Undefined. There is no real number y such that a^y = 0 for any positive base a. The logarithm approaches negative infinity as the argument approaches zero from the right.

Properties of Logarithms

Inverse Relationship with Exponential Functions

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, and asymptotic behavior — determine how logarithms interact with equations, inequalities, and graphs.

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

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

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.