Calculate logarithms of any base. Compute log_b(x) = ln(x)/ln(b) for common, natural, and custom base logarithms. Free online log calculator.
The Logarithm Calculator computes the logarithm of any positive number to any positive base. Logarithms answer the question: "To what power must I raise the base to get x?" If b^y = x, then log_b(x) = y.
This tool supports all common bases: base 10 (common logarithm), base e (natural logarithm, ln), base 2 (binary logarithm), and any custom base. The calculation uses the change of base formula: log_b(x) = ln(x) / ln(b).
Logarithms are essential in science (pH scale, Richter scale, decibels), computer science (binary search complexity, information theory), finance (continuous compounding), and engineering (signal processing).
By calculating this metric accurately, professionals gain actionable insights that support smarter work habits, more realistic scheduling, and improved work-life balance over time. Understanding this metric in precise terms allows professionals to set achievable targets, measure progress objectively, and continuously refine their approach to time and task management.
By calculating this metric accurately, professionals gain actionable insights that support smarter work habits, more realistic scheduling, and improved work-life balance over time.
Logarithms with arbitrary bases are not available on basic calculators. This tool computes any log instantly using the change-of-base formula. Data-driven tracking enables proactive schedule management, helping professionals protect focused work time and reduce the cognitive overhead of constant task-switching throughout the day. This quantitative approach replaces vague time estimates with concrete data, enabling professionals to plan realistic schedules and avoid the pattern of chronic overcommitment.
log_b(x) = ln(x) / ln(b) Where: - b = base (must be > 0 and ≠ 1) - x = the argument (must be > 0) - ln = natural logarithm (base e)
Result: 3
log₁₀(1000) = 3 because 10³ = 1000. Using the formula: ln(1000)/ln(10) = 6.9078/2.3026 = 3.
Many real-world measurements use logarithmic scales because the quantities span many orders of magnitude. The Richter scale for earthquakes, decibels for sound, pH for acidity, and stellar magnitude for brightness all use logarithms.
Continuous compounding uses the natural logarithm: the time to double an investment at rate r is ln(2)/r. The Rule of 72 (72/r%) is a simplified version of this.
log(ab) = log(a) + log(b), log(a/b) = log(a) − log(b), and log(a^n) = n×log(a). These properties simplify complex calculations and form the basis of the slide rule.
Consistent practice with varied problems builds computational fluency and deepens conceptual understanding that transfers across many technical fields.
A logarithm answers: "To what exponent must I raise the base to get this number?" log₁₀(100) = 2 because 10² = 100. It is the inverse of exponentiation.
The natural logarithm (ln) uses base e ≈ 2.71828. It appears naturally in calculus, probability, and continuous growth/decay formulas.
log_b(x) = ln(x)/ln(b) or log_b(x) = log(x)/log(b). This lets you compute any base logarithm using just natural or common logs.
In real numbers, the logarithm is only defined for positive values. There is no real exponent that makes a positive base equal zero or a negative number.
Binary logarithms (log₂) measure algorithm complexity. Binary search takes O(log₂ n) steps. Information entropy is measured in bits using log₂.
They are inverse functions. If y = b^x, then x = log_b(y). This means log_b(b^x) = x and b^(log_b(x)) = x.