Why must the argument of a logarithm be positive?

The logarithm function is defined only for positive inputs. For any base greater than zero and not equal to one, the logarithm of a negative number or zero is undefined. Any candidate solution to a logarithmic inequality must keep every logarithm argument strictly positive; values outside this domain are discarded.

When does the inequality direction flip when converting logarithm to exponential form?

When the base is between 0 and 1, the logarithm is a decreasing function, so the direction flips. When the base is greater than 1, the logarithm is increasing and the direction is preserved. Most textbooks use natural log (base e) or log base 10, both of which preserve direction.

How do you handle a logarithm with a coefficient or added constant?

Isolate the logarithm first. Subtract any added constant. Divide by any coefficient multiplying the logarithm. Once the form is log base b of expression compared to a single number, convert to exponential form. Dividing by a negative coefficient flips the direction once before the conversion.

What does it mean when two logarithms with the same base appear?

If both sides are logarithms with the same base, the comparison reduces to a comparison of their arguments. With base greater than 1 the direction is preserved; with base less than 1 the direction flips. Both arguments must be positive in the final solution.

Free Logarithmic Inequality Solver - Step by Step

Getting Started

Demolog₂(x) > 3, Solve, Clear, the domain at work

Step 0 of 4

The solver shows an inequality display at the top with a blinking yellow caret marking the cursor position. Click anywhere in the display to place the cursor, or use arrow keys, Home, and End for keyboard navigation. Type directly or use the button panel below.

Three quick experiments:

• Type

\log_2(x) > 3

and press Enter — the right panel converts to exponential form

x > 2^3 = 8

, applies the domain

x > 0

(automatically satisfied since

8 > 0

), and reports

x > 8

.
• Click the "Linear Argument" example template — an inequality like

\log_2(2x + 1) > 3

loads. The solver converts to

2x + 1 > 8

, solves the linear inequality, then intersects with the domain

2x + 1 > 0

.
• Try the "Negative Right Side" template — an inequality such as

\log_2(x) > -3

. Converting gives

x > 2^{-3} = 1/8

, automatically positive so the domain is satisfied.

The Solve button is disabled until you enter something. The dedicated "Log" row inserts

\ln

,

\log

(base 10),

\log_2

,

\log_3

, and

\log_5

as single button clicks. An expression without an inequality symbol is refused with "Input must be an inequality (use <, >, ≤, or ≥)", one without a logarithm of the variable with "No logarithm with variable found in inequality", and different bases on the two sides are not yet supported.

\log_2(x) < 3

converts to

x < 8

and, with the domain

x > 0

, gives

(0, 8)

. What makes an inequality logarithmic is explained in what is a logarithmic inequality.

Building Inequalities with Buttons and Keyboard

DemoA linear argument, undo, delete, Solve

Step 0 of 4

The button panel groups inputs by type. All inputs can be made either by clicking or typing on the keyboard.

• Log row — one click inserts the logarithm function with the appropriate base:

\ln

(natural log),

\log

(base 10),

\log_2

,

\log_3

,

\log_5

. After insertion the cursor sits inside the parentheses, ready for the argument.
• Variable row —

x

,

y

,

n

.
• Number row — digits 0 through 9 and the decimal point.
• Operator row — plus, minus, multiplication, division, caret.
• Ineq row — the four inequality symbols.
• Special row —

e

for the natural exponential base, and parentheses.

For non-standard bases, type

\log

then underscore then the base number then open parenthesis:

\log\_7(x)

for

\log_7(x)

. The parser recognizes both the underscore-base form and the bare

\log

(default base 10) and

\ln

(base

e

) forms.

Keyboard shortcuts: type letters and digits directly, use star or slash for multiplication and division, type

<

and

>

directly, type

<=

or

>=

to insert

\leq

and

\geq

, and press Enter to solve. Ctrl+Z undoes up to fifty edits back. The Clear button empties the display entirely; the curved-arrow button steps through the undo stack one edit at a time; the ⌫ button deletes the character before the caret.

\log_2(2x + 1) > 3

is typed by clicking the base button, then

2x + 1

and the closing bracket; it converts to

2x + 1 > 8

and gives

x > 3.5

. Why the domain must be checked is explained in domain and direction.

Try an Example — Eight Form Templates

DemoOpen the templates, load two forms

Step 0 of 4

Click the "Try an Example" header to expand the template panel. Each card generates a random inequality of that form. Clicking again produces a new random version.

• Simple —

\log_b(x) > c

with

b \in \{2, 3, 5, 10\}

. The base case with a single-variable argument.
• Natural Log —

\ln(x) > c

. Uses the natural base

e

, which produces clean exponential-form right sides.
• With Coefficient —

a \cdot \log_b(x) > c

. Requires dividing by

a

to isolate the logarithm before converting.
• Linear Argument —

\log_b(mx + n) > c

. The logarithm argument is itself a linear expression; the post-conversion linear inequality is non-trivial.
• With Constant —

\log_b(x) + c > d

. Requires subtracting

c

to isolate the logarithm.
• Natural Linear —

\ln(mx + n) > c

. Combines the natural base with a linear argument.
• Same Base Both Sides —

\log_b(f(x)) > \log_b(c)

. Exercises the same-base comparison shortcut: the arguments are compared directly.
• Negative Right Side —

\log_b(x) > -c

. Tests the case where the exponential value

b^{-c}

is a small positive number rather than zero or negative.

Roughly 80 percent of generated inequalities use clean bases and right-hand sides that produce exact integer or rational boundary values. Loading a template clears the previous solution; click Solve to work the new inequality. The five stages applied to every template are listed in the solving process explained.

Reading the Step-by-Step Solution

DemoA coefficient to divide out, then the natural log

Step 0 of 3

The solution panel shows each algebraic move as a labeled step. The exact sequence depends on the structural case.

• Rearrange Inequality (when applicable) — if the logarithm is on the right, the solver swaps sides and flips the direction.
• Evaluate Constant — simplifies the non-logarithm side to a single number.
• Isolate Logarithm — subtracts additive constants or divides by a coefficient outside the logarithm to get the form

\log_b(\text{expr}) \square c

.
• Compare Logarithms (for same-base case) — if both sides are logarithms with the same base, the solver removes the logarithm from both sides and continues with the arguments. Direction is preserved when the base is greater than 1 and flipped when less than 1.
• Convert to Exponential Form — restates

\log_b(A) \square c

as

A \square b^c

. Direction flips when the base is less than 1.
• Evaluate Exponential — computes

b^c

to a single number.
• Domain Constraint — reminds that the original argument must be positive for the logarithm to be defined.
• Solve Linear Inequality — finishes the post-conversion linear inequality in the variable.
• Apply Domain Constraint — intersects the algebraic solution with the requirement that the argument be positive. May produce a bounded interval rather than a single ray.

The final-answer card shows the result as an inequality form plus interval notation.

2 \cdot \log(x) \leq 4

is divided by

2

first, converts to

x \leq 100

and, with the domain, gives

(0, 100]

;

\ln(x) \geq 0

gives

x \geq 1

, the interval

[1, \infty)

. The box under the solver, How the solver works, summarises the method in a few lines and links to the theory below. The two things that can go wrong are explained in domain and direction.

Domain and Direction — The Two Hazards

Logarithmic inequality solving has two distinct hazards. Tracking both is critical.

Hazard 1: Domain restriction. A logarithm is defined only when its argument is positive. So

\log_b(f(x))

is defined when

f(x) > 0

. Any candidate value of

x

that violates this is discarded, regardless of what the algebraic solution says.

• For an isolated

\log_b(x) \square c

, the post-conversion inequality

x \square b^c

may include values where

x \leq 0

that must be excluded.
• For

\log_b(\text{linear expression})

, the argument must be positive:

mx + n > 0

adds its own boundary to combine with the algebraic answer.
• The final answer is always the intersection: the algebraic solution AND the domain.

Hazard 2: Direction flip with bases less than 1. The logarithm function

\log_b

is increasing when

b > 1

and decreasing when

0 < b < 1

. Converting between logarithmic and exponential forms with a base less than 1 reverses the inequality direction.

•

\log_{1/2}(x) > 3

converts to

x < (1/2)^3 = 1/8

(direction flipped).
•

\log_2(x) > 3

converts to

x > 2^3 = 8

(direction preserved).

The combination. When both hazards apply, the solver handles them sequentially: first the algebraic conversion with the appropriate direction, then the domain intersection. The result is either a single ray, a bounded interval, or no solution.

Examples.

•

\log_2(x) < 3

becomes

x < 8

with domain

x > 0

. Intersection:

0 < x < 8

, a bounded interval.
•

\log_2(x) > -3

becomes

x > 1/8

with domain

x > 0

. Domain is automatic since

1/8 > 0

. Final answer:

x > 1/8

.
•

\log_2(x) < -3

becomes

x < 1/8

with domain

x > 0

. Intersection:

0 < x < 1/8

.

For deeper coverage see logarithm and domain of a function.

The domain hazard is treated in domain considerations; the direction rule in base greater than one: direction preserved and base between zero and one: direction reversed.

What is a Logarithmic Inequality?

A logarithmic inequality is an inequality where the variable appears inside a logarithm. The standard form is

\log_b(f(x)) \;\square\; c

where

b > 0

and

b \neq 1

is the base,

f(x)

is a function of the variable (usually linear),

c

is a real number, and

\square

is one of

<

,

>

,

\leq

,

\geq

.

The two regimes by base.

• $b > 1$ :

\log_b

is strictly increasing. Larger argument means larger logarithm. Converting between logarithmic and exponential form preserves direction.
• $0 < b < 1$ :

\log_b

is strictly decreasing. Larger argument means smaller logarithm. Conversion reverses direction.

The domain restriction. For any base,

\log_b(f(x))

requires

f(x) > 0

. This restriction must be enforced separately and intersected with the algebraic solution.

Solution shapes. Depending on the structural case, a logarithmic inequality can produce:

• A single ray (most common when the algebraic solution agrees with the domain).
• A bounded interval (when the algebraic solution caps from above and the domain caps from below).
• A half-line starting at the domain boundary.
• No solution (when the algebraic solution is disjoint from the domain).

The pivotal move. Converting

\log_b(f(x)) \square c

to

f(x) \square b^c

extracts the variable from the logarithm. From there, standard linear (or polynomial) inequality techniques finish the problem.

Same-base shortcut. When both sides are logarithms with the same base, the conversion can be skipped:

\log_b(A) \square \log_b(B)

becomes

A \square B

directly, with direction reversed for bases less than 1. Both arguments must still be positive.

For deeper coverage see logarithmic inequality, logarithm, and exponential function.

The place of these inequalities among the logarithm topics is logarithmic inequalities, the picture behind the solution shapes is graphical interpretation, and the allowed inputs are restrictions on base and argument.

The Solving Process Explained

Five stages handle any standard logarithmic inequality.

• Stage 1: Isolate the logarithm. Move the logarithm to one side. Subtract any constants added to it. Divide by any coefficient multiplying it (flipping direction if the coefficient is negative). The goal is to reach

\log_b(f(x)) \square c

where the right side is a single number.

• Stage 2: Recognize same-base shortcut. If both sides are logarithms with the same base after isolation, skip the exponential conversion and compare arguments directly:

\log_b(A) \square \log_b(B)

becomes

A \square B

(or flipped if

b < 1

). Both

A > 0

and

B > 0

must hold.

• Stage 3: Convert to exponential form. Replace

\log_b(f(x)) \square c

with

f(x) \square b^c

. Direction flips if and only if

b < 1

.

• Stage 4: Solve the resulting inequality. Compute

b^c

as a number. The inequality is now

f(x) \square b^c

. If

f(x)

is linear, the standard linear-inequality moves finish it.

• Stage 5: Intersect with the domain. The original argument

f(x)

must be positive. Solve

f(x) > 0

and intersect with the result of stage 4. The intersection is the final answer.

The implementation. Internally, the solver navigates this decision tree based on the structure of the input AST. It detects same-base shortcuts, applies appropriate direction-flip logic per base, and assembles the final answer by intersecting the algebraic solution with the domain. The result is rendered as an inequality plus the corresponding interval notation.

Edge cases the solver handles automatically.

• No-solution after intersection — when the algebraic answer and the domain do not overlap.
• Domain automatic — when

b^c > 0

trivially satisfies the domain (which is always true since

b > 0

).
• Bounded interval — when the algebraic answer caps from above and the domain caps from below (or vice versa).

For comprehensive treatment see solving logarithmic inequalities and logarithm rules.

The same-base shortcut is logarithms on both sides, merging several logarithms first is inequalities with combined logarithms, and the whole procedure is summarised in summary of techniques.

Related Concepts

Logarithm — the inverse of the exponential function.

\log_b c

is the unique exponent

k

such that

b^k = c

. The pivotal tool that this solver works with.

Natural Logarithm — logarithm with base

e \approx 2.71828

. Denoted

\ln

. Appears throughout calculus, probability, and information theory.

Common Logarithm — logarithm with base 10. Denoted

\log

without a subscript in most contexts. Standard for decibels, pH, and Richter scale.

Logarithm Base 2 — appears in computer science and information theory. Denoted

\log_2

.

Exponential Function — the inverse of the logarithm.

b^x

where

b > 0, b \neq 1

. Logarithmic inequalities reduce to exponential equations via conversion.

Exponential Inequality — the conjugate problem type. An exponential inequality with the variable in the exponent is solved by taking a logarithm; a logarithmic inequality with the variable inside the logarithm is solved by exponentiating.

Domain of a Function — the set of inputs for which the function is defined. For logarithms, the domain is the positive real numbers; for the argument to be a valid input, it must evaluate to something positive.

Change of Base — the formula

\log_b c = \frac{\log c}{\log b} = \frac{\ln c}{\ln b}

. Converts any logarithm to a more convenient base for computation.

Logarithm Rules — the algebraic identities including product, quotient, and power rules. Useful for combining or splitting logarithms before isolation:

\log(ab) = \log a + \log b

,

\log(a/b) = \log a - \log b

,

\log(a^k) = k \log a

.

Linear Inequality — the inequality type that the post-conversion step usually produces. The solver routes through linear-inequality logic to finish.

Compound Inequality — the form like

a < x \leq b

that arises when intersecting an upper bound from the algebraic solution with a lower bound from the domain.

pH — the canonical real-world logarithmic quantity.

\text{pH} = -\log_{10}[\text{H}^+]

. Inequalities about pH ranges reduce to logarithmic inequalities.

Decibel — another logarithmic quantity. Sound levels measured in decibels follow logarithm rules; threshold-comparison problems are logarithmic inequalities.

Interval Notation — the standard way to write the solution. Parentheses for excluded endpoints, brackets for included endpoints, infinity always with parentheses.

The related tools are the logarithmic equation solver, the exponential inequality solver and the logarithm calculator; the lessons are compound inequalities, logarithm rules and common and natural logarithms.