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 log2(x)>3 and press Enter — the right panel converts to exponential form x>23=8, applies the domain x>0 (automatically satisfied since 8>0), and reports x>8. • Click the "Linear Argument" example template — an inequality like log2(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 log2(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), log2, log3, and log5 as single button clicks.
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.
• Log row — one click inserts the logarithm function with the appropriate base: ln (natural log), log (base 10), log2, log3, log5. 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 log7(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 ≤ and ≥, and press Enter to solve. 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 — logb(x)>c with b∈{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⋅logb(x)>c. Requires dividing by a to isolate the logarithm before converting. • Linear Argument — logb(mx+n)>c. The logarithm argument is itself a linear expression; the post-conversion linear inequality is non-trivial. • With Constant — logb(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 — logb(f(x))>logb(c). Exercises the same-base comparison shortcut: the arguments are compared directly. • Negative Right Side — logb(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.
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 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 logb(expr)□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 logb(A)□c as A□bc. Direction flips when the base is less than 1. • Evaluate Exponential — computes bc 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.
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 logb(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 logb(x)□c, the post-conversion inequality x□bc may include values where x≤0 that must be excluded. • For logb(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 logb 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.
• log1/2(x)>3 converts to x<(1/2)3=1/8 (direction flipped). • log2(x)>3 converts to x>23=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.
• log2(x)<3 becomes x<8 with domain x>0. Intersection: 0<x<8, a bounded interval. • log2(x)>−3 becomes x>1/8 with domain x>0. Domain is automatic since 1/8>0. Final answer: x>1/8. • log2(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.
What is a Logarithmic Inequality?
A logarithmic inequality is an inequality where the variable appears inside a logarithm. The standard form is
logb(f(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.
• $b > 1$: logb is strictly increasing. Larger argument means larger logarithm. Converting between logarithmic and exponential form preserves direction. • $0 < b < 1$: logb is strictly decreasing. Larger argument means smaller logarithm. Conversion reverses direction.
The domain restriction. For any base, logb(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 logb(f(x))□c to f(x)□bc 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: logb(A)□logb(B) becomes A□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 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 logb(f(x))□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: logb(A)□logb(B) becomes A□B (or flipped if b<1). Both A>0 and B>0 must hold.
• Stage 3: Convert to exponential form. Replace logb(f(x))□c with f(x)□bc. Direction flips if and only if b<1.
• Stage 4: Solve the resulting inequality. Compute bc as a number. The inequality is now f(x)□bc. 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 bc>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.
Related Concepts
Logarithm — the inverse of the exponential function. logbc is the unique exponent k such that bk=c. The pivotal tool that this solver works with.
Natural Logarithm — logarithm with base e≈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 log2.
Exponential Function — the inverse of the logarithm. bx where b>0,b=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 logbc=logblogc=lnblnc. 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)=loga+logb, log(a/b)=loga−logb, log(ak)=kloga.
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≤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. pH=−log10[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.