When does the inequality direction flip in exponential solving?

When the base is between 0 and 1, the corresponding logarithm is a decreasing function, so taking the logarithm of both sides flips the inequality direction. When the base is greater than 1, the logarithm is increasing and direction is preserved. The solver tracks this automatically based on the base.

What if the exponential is greater than a negative number?

Exponential expressions with a positive base are always positive: any base to any real power is greater than zero. So an inequality of the form b to the x is greater than a negative number is automatically true for every real x. Conversely, b to the x is less than a negative number has no solution.

How do you handle coefficients and additive constants outside the exponential?

Isolate the exponential first. Subtract any constant that is added to or subtracted from the exponential. Then divide by any coefficient multiplying the exponential. Once the form is b to the x compared to c, apply the logarithm. Dividing by a negative coefficient flips the direction once before the logarithm is applied.

When should you use the natural logarithm versus log base b?

Both work; the choice affects only how the answer is written. For the base e (natural exponential), the natural log gives a clean answer. For another base b, you can use log base b directly or use the change-of-base formula to express the answer in terms of natural log or common log.

Exponential Inequality Solver

How to use Exponential Inequality Solver

Build the inequality in the display at the top: click the Var, Num and Op buttons, with ^ for the exponent and e for the natural base, one of the four Ineq symbols and the brackets, or type on the keyboard; wrap a multi-term exponent in brackets. Learn more about building inequalities
Clear empties the display, the curved arrow undoes the last edit and the ⌫ button deletes the character before the caret. Learn more about editing
Click Solve, or press Enter in the display, once the inequality is complete; an expression without an inequality symbol or without a variable in an exponent is refused. Learn more about solving
Open Try an Example and click a card to load a random inequality of that form. Learn more about the templates
The right panel lists the steps from isolating the exponential to the logarithm and the linear inequality, with a separate step whenever the sign flips, and the answer card gives the inequality and the interval; the box under the solver explains the method. Learn more about the solution
An exponential compared with zero or a negative number ends in All real numbers or No solution without any logarithm. Learn more about these cases

Solve inequalities with variables in exponents

Enter inequality...

Var

Num

Ineq

Select an inequality type or enter your own, then click Solve to see the step-by-step solution.

How the solver works

The solver reads one inequality in

x

y

n

with the variable in an exponent, such as

a cdot b^{f(x)}

compared with

c

for a positive base $b
eq 1

. It isolates the exponential, flipping the sign if it divides by a negative coefficient, settles the comparisons with zero or a negative number at once, then takes the logarithm of both sides, flipping the sign again when the base is below

1$, and solves the linear inequality that remains. A right side that is a whole power of the base gives an exact boundary. Learn more about exponential inequalities

Getting Started

Building Inequalities with Buttons and Keyboard

Try an Example — Eight Form Templates

Reading the Step-by-Step Solution

The Two Direction-Flip Triggers

What is an Exponential Inequality?

The Solving Process Explained

Related Concepts

Getting Started

Demo2^x > 8, Solve, Clear, no solution

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

2^x > 8

and press Enter — the right panel detects

8 = 2^3

exactly, applies log base 2, and reports

x > 3

as an exact answer.
• Click the "Base less than 1" example template — an inequality with base

0.5

loads. Watch the solver flip the inequality direction when applying the logarithm because the log of a decreasing exponential reverses comparisons.
• Try the "Compare to Zero" template to see how the solver handles a positive base raised to any power being compared with zero or a negative number — one direction is always-true, the other has no solution.

The Solve button is disabled until you enter something. Pressing Enter on the display is equivalent to clicking Solve. The dedicated "Ineq" row inserts the four inequality symbols. The

e

button inserts the natural exponential base. An expression without an inequality symbol is refused with "Input must be an inequality (use <, >, ≤, or ≥)", one without a variable in an exponent with "No variable found in any exponent", and exponentials on both sides are not yet supported.

2^x < 0

ends in No solution before any logarithm. What makes an inequality exponential is explained in what is an exponential inequality.

Building Inequalities with Buttons and Keyboard

DemoA base below 1, undo, delete, the flip

Step 0 of 4

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

• Variable row —

x

y

n

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

e

for the natural exponential base, and parentheses.

The caret operator is the key tool for this solver: typing

2

then caret then

x

creates

2^x

, which the display renders with the exponent as a superscript. For multi-term exponents use parentheses:

2^{(3x + 1)}

requires parentheses around the exponent to bind it correctly.

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

<

and

>

directly, type

<=

>=

to insert

\leq

and

\geq

, and press Enter to solve. 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.

0.5^x < 0.125

is typed with the decimal point; its logarithm step flips the sign and gives

x > 3

. Why a base below

1

flips the sign is explained in the two direction-flip triggers. Type

e

for Euler's number; the parser recognizes it as the natural base only when not followed by another letter. Ctrl+Z undoes up to fifty edits back.

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 ( $>$ ) —

b^x > c

or one of the other directions. The base case with a single-variable exponent.
• With Coefficient —

a \cdot b^x > c

. Requires dividing by

a

before applying the logarithm.
• Linear Exponent —

b^{mx + n} > c

. The exponent is itself a linear expression in the variable; after the logarithm, a linear inequality must be solved.
• Base less than 1 (Flips!) —

(1/2)^x > c

or similar. Exercises the direction-flip branch because the log of a base less than 1 reverses the comparison.
• With Constant —

b^x + c > d

. Requires subtracting

c

to isolate the exponential.
• Natural Base —

e^x > c

. The natural exponential function; uses

\ln

directly for a clean answer.
• Compare to Zero —

b^x > 0

b^x <

negative. Exercises the always-true and no-solution edge cases.
• Natural Linear —

e^{mx + n} > c

. Combines the natural base with a linear exponent.

Roughly 80 percent of generated inequalities use clean bases and right-hand sides that produce exact integer or rational answers. 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 linear exponent, then the natural base

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 exponential is on the right, the solver swaps sides and flips the direction.
• Evaluate Constant — simplifies the non-exponential side to a single number.
• Isolate Exponential Term — subtracts additive constants or divides by a coefficient outside the exponential to get the form

b^{\text{expr}} \square c

.
• Exponential Properties (for special cases) — handles right side at or below zero. Returns all-reals or no-solution without applying the logarithm.
• Apply Logarithm (Base $< 1$ ) — explicitly notes the inequality sign flips because the logarithm base less than 1 is decreasing.
• Apply Logarithm — standard logarithm, direction preserved because the base is greater than 1.
• Simplify — if

c = b^k

for some integer

k

, the logarithm evaluates exactly to

k

; otherwise the solver reports a decimal approximation.
• Simplify Logarithm (when exponent is linear) — restates the post-logarithm inequality in terms of the original variable.
• Solve Linear Inequality — solves for the variable using the standard linear-inequality moves. May trigger another direction flip if the variable coefficient inside the exponent is negative.

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

3^{2x - 1} \leq 27

recognises

27 = 3^3

, reduces to

2x - 1 \leq 3

and gives

x \leq 2

, the interval

(-\infty, 2]

;

e^x \geq 1

gives

x \geq 0

. The box under the solver, How the solver works, summarises the method in a few lines and links to the theory below. Where the sign can flip is explained in the two direction-flip triggers.

The Two Direction-Flip Triggers

Exponential inequality solving has two distinct places where the inequality direction can flip. Tracking both is critical.

Trigger 1: Dividing by a negative coefficient during isolation. If the inequality is

-3 \cdot 2^x > 24

, dividing by

-3

gives

2^x < -8

. The flip is the standard linear-inequality flip, applied to the equivalent inequality between the exponential and a constant. After this point the inequality may have an impossible right side (negative), triggering the no-solution special case.

Trigger 2: Applying a logarithm with base less than 1. The function

\log_b

is increasing when

b > 1

and decreasing when

0 < b < 1

. Applying a decreasing function to both sides of an inequality reverses the direction. So an inequality like

(1/2)^x > 8

becomes

x \log(1/2) > \log 8

, and dividing by the negative

\log(1/2)

flips again — net effect: one flip.

Equivalent perspective. Rewriting

(1/2)^x = 2^{-x}

transforms

(1/2)^x > 8

into

2^{-x} > 2^3

, which gives

-x > 3

and finally

x < -3

. The direction flip arises whether you handle it through the logarithm rule or through the negation in the exponent — same answer either way.

The solver's approach. The solver detects whether the base is less than 1 and inserts an explicit flip step labeled "Apply Logarithm (Base

< 1

)". For bases greater than 1 the corresponding step is labeled simply "Apply Logarithm" with no direction change.

Cumulative flips. If both triggers fire (negative coefficient outside the exponential, base less than 1), the two flips cancel and the original direction is restored. The solver tracks them step by step rather than computing parity.

For deeper coverage see exponential function and logarithm.

The base rule is the subject of key principle: base determines direction; the sign rules for the other moves are properties of inequalities, and the logarithm as the inverse of the exponential is inverse identities.

What is an Exponential Inequality?

An exponential inequality is an inequality where the variable appears in an exponent. The standard form is

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.

• $b > 1$ : the exponential function

b^x

is strictly increasing. Larger

x

means larger

b^x

. Inequalities of the form

b^x > c

correspond to

x >

something.
• $0 < b < 1$ : the exponential function

b^x

is strictly decreasing. Larger

x

means smaller

b^x

. Inequalities reverse direction when extracting the variable:

b^x > c

corresponds to

x <

something.

The pivotal observation. Since

b^x > 0

for every real

x

, comparisons of

b^{f(x)}

to non-positive values are trivial:

•

b^{f(x)} >

(non-positive number): always true.
•

b^{f(x)} \geq

(non-positive number): always true.
•

b^{f(x)} <

(non-positive number): no solution.
•

b^{f(x)} \leq

(non-positive number): no solution.

These cases are dispatched immediately without applying any logarithm.

Solution form. Once isolated to

b^{f(x)} \square c

with

c > 0

, the inequality reduces to

f(x) \square \log_b c

(with possible direction flip for

b < 1

). The result is then a linear inequality in

x

, whose solution is a ray on the number line.

For deeper coverage see exponential function, exponential equation, and logarithm.

The basic form is solved in solving basic exponential inequalities, the comparisons with zero in domain and sign considerations, and the place of these inequalities among the power topics in exponential inequalities.

The Solving Process Explained

Five stages handle any standard exponential inequality.

• Stage 1: Isolate the exponential. Move the exponential 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

b^{f(x)} \square c

where

b

is a positive base not equal to 1 and

c

is a constant.

• Stage 2: Examine the special cases.

• $c \leq 0$ with greater-than form: always true. Solution is all reals.
• $c \leq 0$ with less-than form: no solution.
• Otherwise proceed.

• Stage 3: Apply the logarithm. Take

\log_b

(or any logarithm, with adjustment) of both sides. The inequality becomes

f(x) \square \log_b c

. If the base

b

is less than 1, the direction flips at this step because

\log_b

is decreasing.

• Stage 4: Detect perfect powers. If

c = b^k

for some integer

k

, the logarithm evaluates exactly to

k

and the answer is clean. Otherwise the logarithm value is a decimal approximation.

• Stage 5: Solve the resulting linear inequality. If

f(x)

is linear, the inequality

f(x) \square \log_b c

is a standard linear inequality, solved by the usual moves (subtract constants, divide by coefficient with possible direction flip if the coefficient is negative).

The implementation. Internally, the solver navigates this five-stage process based on the structure of the input AST. It detects base regime, applies special-case shortcuts where possible, and assembles the answer as an inequality of the form

x \square v

with the matching interval notation.

For comprehensive treatment see solving exponential inequalities and logarithms.

The isolation stage is worked in inequalities requiring simplification, the logarithm stage in method: using logarithms, and the cases are summarised in summary of cases.