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 2x>8 and press Enter — the right panel detects 8=23 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.
Building Inequalities with Buttons and Keyboard
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 2x, 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 <= or >= to insert ≤ and ≥, and press Enter to solve. 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
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 ($>$) — bx>c or one of the other directions. The base case with a single-variable exponent. • With Coefficient — a⋅bx>c. Requires dividing by a before applying the logarithm. • Linear Exponent — bmx+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 — bx+c>d. Requires subtracting c to isolate the exponential. • Natural Base — ex>c. The natural exponential function; uses ln directly for a clean answer. • Compare to Zero — bx>0 or bx< negative. Exercises the always-true and no-solution edge cases. • Natural Linear — emx+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.
Reading the Step-by-Step Solution
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 bexpr□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=bk 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.
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⋅2x>24, dividing by −3 gives 2x<−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 logb 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 xlog(1/2)>log8, 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>23, 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.
What is an Exponential Inequality?
An exponential inequality is an inequality where the variable appears in an exponent. The standard form is
bf(x)□c
where b>0 and b=1 is the base, f(x) is a function of the variable (usually linear), c is a real number, and □ is one of <, >, ≤, ≥.
The two regimes by base.
• $b > 1$: the exponential function bx is strictly increasing. Larger x means larger bx. Inequalities of the form bx>c correspond to x> something. • $0 < b < 1$: the exponential function bx is strictly decreasing. Larger x means smaller bx. Inequalities reverse direction when extracting the variable: bx>c corresponds to x< something.
The pivotal observation. Since bx>0 for every real x, comparisons of bf(x) to non-positive values are trivial:
These cases are dispatched immediately without applying any logarithm.
Solution form. Once isolated to bf(x)□c with c>0, the inequality reduces to f(x)□logbc (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 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 bf(x)□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 logb (or any logarithm, with adjustment) of both sides. The inequality becomes f(x)□logbc. If the base b is less than 1, the direction flips at this step because logb is decreasing.
• Stage 4: Detect perfect powers. If c=bk 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)□logbc 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□v with the matching interval notation.
For comprehensive treatment see solving exponential inequalities and logarithms.
Related Concepts
Exponential Function — the function f(x)=bx where b>0, b=1. Strictly increasing for b>1, strictly decreasing for 0<b<1, always positive.
Exponential Equation — the equality counterpart bf(x)=c. Solving uses the same logarithm step but produces a single equation in x rather than an inequality.
Logarithm — the inverse of the exponential. logbc is the unique exponent k such that bk=c. The pivotal tool for extracting the variable from the exponent in exponential inequalities.
Natural Logarithm — logarithm with base e. Denoted ln. The default choice for inequalities involving the natural exponential ex.
Common Logarithm — logarithm with base 10. Denoted log (without a subscript) in many contexts. Useful when the base is 10 or when expressing approximations.
Change of Base — the formula logbc=logblogc=lnblnc. Converts any logarithm to one with a more convenient base, which is how non-standard bases are computed.
Linear Inequality — the inequality type that the post-logarithm step produces. The solver routes through the linear-inequality logic to finish.
Compound Interest — the real-world application driving most exponential inequalities. Questions like "how long until the balance exceeds X" reduce to exponential inequalities in time.
Half-Life and Decay — the decreasing-exponential context. Quantities like radioactive material or drug concentration follow A0⋅(1/2)t/T, and questions about when concentration drops below a threshold are exponential inequalities with base less than 1.
e (Euler's Number) — the natural exponential base, approximately 2.71828. Appears in continuous compounding, calculus, probability, and complex analysis.
Interval Notation — the standard way to write the solution as an open or closed ray. Parentheses for excluded endpoints (strict inequalities), brackets for included endpoints (non-strict), infinity always with parentheses.