HOW TO calculate natural log (ln)?

Set base to e using the quick button or type 2.718281828. The calculator uses ln(x)=log_e(x) and shows the change-of-base derivation.

HOW TO compute common log (log10)?

Use base=10. The tool displays log10(x) and provides step-by-step change-of-base to verify results.

HOW TO handle invalid inputs?

Logs are defined only for positive numbers and bases greater than zero but not equal to 1. The calculator provides clear error messages to guide you.

HOW TO calculate natural log (ln)?

Set base to e using the quick button or type 2.718281828. The calculator uses ln(x)=log_e(x) and shows the change-of-base derivation.

HOW TO compute common log (log10)?

Use base=10. The tool displays log10(x) and provides step-by-step change-of-base to verify results.

HOW TO handle invalid inputs?

Logs are defined only for positive numbers and bases greater than zero but not equal to 1. The calculator provides clear error messages to guide you.

Log Calculator | TheCalculatorToolbox

Log Calculator | Fast, Accurate, Mobile-Friendly Logarithm Solver

Enter values

Number x

x

x must be positive (x > 0)

Base b

b

b > 0 and b ≠ 1

Use quick base buttons for ln, log10, and log2.

Understand logarithms

A logarithm answers: “to what exponent must the base be raised to get x?” Formally, log_b(x) = y means b^y = x, where x > 0, b > 0, and b ≠ 1. This calculator computes log values for any valid base, with clear steps via the change-of-base formula.

Change-of-base formula

The most reliable way to compute logs on computers is: log_b(x) = ln(x) / ln(b) = log₁₀(x) / log₁₀(b). This ensures consistency across different bases and is used under the hood for accuracy.

Key properties

Product rule: log_b(xy) = log_b(x) + log_b(y)

Quotient rule: log_b(x/y) = log_b(x) − log_b(y)

Power rule: log_b(x^k) = k · log_b(x)

Base switch: log_b(x) = ln(x)/ln(b) = log₁₀(x)/log₁₀(b)

Domain and special cases

Domain: x must be positive (x > 0). Base must be positive and not equal to 1.

ln(x): natural log uses base e ≈ 2.718281828.

log₁₀(x): common log is widely used in scientific notation and pH scales.

log₂(x): binary log helps estimate bit-length and algorithmic complexity.

Who benefits

Students and educators for algebra and calculus; engineers for signal scales (dB) and materials testing; data scientists for feature scaling; software developers for complexity analysis; finance professionals for compound growth modeling. The tool is global and language-agnostic—no regional formats required.

How to read the results

Value: The computed log_b(x) with precision handling.

Change-of-base: Shows ln(x)/ln(b) so you can verify the calculation steps.

Quick comparisons: Also displays ln(x), log₁₀(x), and log₂(x) for context.

Warnings: Clear messages when inputs violate the domain.

FAQs

HOW TO use the log calculator?

Enter x and base b, then select Calculate. You’ll see log_b(x), change-of-base steps, and quick references for ln, log10, and log2.

HOW TO compute natural log (ln)?

Select “Set base e” or type 2.718281828 in the base field and calculate.

HOW TO calculate common log (log10)?

Use base b = 10 for scientific and engineering applications.

HOW TO find binary log (log2)?

Use base b = 2 to estimate bits and complexity in computer science tasks.

HOW TO fix invalid input errors?

Ensure x > 0 and b > 0 with b ≠ 1. The tool prevents undefined results and guides with messages.

HOW TO improve precision?

Enter values with full precision and review exponential-form outputs for very large or small numbers.

HOW TO use this globally?

All computation is client-side and country-neutral, using universal math notation.

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