Logarithmic inequality solver

It supports natural logarithms, common logarithms, and explicit positive bases other than one. The calculation also accounts for the crucial direction change when the base lies between zero and one, so the result includes both the transformed inequality and the domain intersection instead of treating the logarithm like an unrestricted algebraic expression.

Enter a supported logarithmic inequality

Write the logarithm on the left and a finite numeric constant on the right. Accepted forms include ln(2x + 1) <= 0, log(x) > 2, log10(x - 4) >= 1, and log_2(x - 1) > 3. The unmarked log form means base ten, while ln means base e. An explicit base follows the underscore and must be positive but cannot equal one. Inside the logarithm, the solver accepts one linear x term plus an optional constant, with or without an explicit multiplication sign. This focused grammar keeps the result transparent and deterministic: each accepted expression reduces to two linear inequalities, rather than being passed to an opaque symbolic engine. The right side must be a decimal number, and the comparison must be one of the strict or inclusive operators. Expressions with powers of x, products of variables, multiple logarithms, variables on the right, or compound comparisons are rejected with an input error instead of producing a plausible but unsupported answer. That boundary makes the tool suitable for lessons, answer checking, and software tests where a clearly defined contract matters as much as the result.

Convert to exponential form without losing the direction

For a base greater than one, the logarithm is increasing. Therefore log base b of an argument being greater than c is equivalent to the argument being greater than b raised to c, and the same comparison sign is retained. A base between zero and one behaves differently because its logarithm is decreasing. In that case every comparison reverses during conversion: log base b of an argument greater than c becomes the argument less than b raised to c. The response exposes the base, monotonicity, exponential form, evaluated threshold, and transformed linear inequality so this decision is visible. For example, log base one half of x greater than one converts to x less than one half, not x greater than one half. Inclusive operators remain inclusive after any required reversal. The calculation uses ordinary finite real arithmetic and rejects thresholds that overflow to infinity or underflow to zero, since emitting a rounded finite boundary in those cases would misrepresent the solution. No network, random value, current date, or remote computer algebra service participates, so identical valid input always produces identical structured output and step text.

Respect the domain and read the interval

A real logarithm exists only when its argument is strictly positive. That restriction is calculated before the exponential comparison and then intersected with the inequality obtained from conversion. This matters most when the converted condition points toward values that cross the argument's zero. For ln(2x + 1) less than or equal to zero, conversion gives 2x + 1 less than or equal to one, but the domain also requires 2x + 1 greater than zero. Combining them yields an interval that is open at negative one half and closed at zero. The response reports the domain separately, followed by a plain solution and interval notation. Parentheses mark excluded endpoints, square brackets mark included finite endpoints, and infinity always uses a parenthesis. A one-sided result such as x greater than nine appears as (9, ∞), while a bounded result can appear as (-0.5, 0]. The steps array is intended for explanation and auditing: it states the positivity condition, identifies whether the logarithm increases or decreases, shows the exponential conversion, and records the final intersection. The solver does not silently discard the domain even when the converted inequality already implies positivity, because showing it is part of solving a logarithmic inequality correctly.