Logarithm Calculator
Calculate log base 10, natural log (ln), log base 2, or any custom base — with instant results and step-by-step breakdown
Calculate Logarithm
Must be a positive number greater than 0
Must be positive and not equal to 1
Result
How to Use the Logarithm Calculator
Key Features
Common Log (Base 10)
Calculate log base 10 used in science, engineering, decibel scales, and pH calculations.
Natural Log (Base e)
Compute ln used in calculus, continuous growth models, and advanced mathematics.
Binary Log (Base 2)
Find log base 2 essential for computer science, algorithm analysis, and information theory.
Custom Base
Enter any positive base (except 1) for specialized calculations in any mathematical context.
Step-by-Step Breakdown
See the formula, change of base conversion, and antilog verification for every calculation.
Instant & Precise
Results computed to 10 decimal places using 64-bit double precision floating-point math.
How Logarithms Work — The Formula
b^y = x, then log_b(x) = y. For example, log₁₀(1000) = 3 because 10³ = 1000.log_b(x) = ln(x) / ln(b). This formula allows computing logarithms of any valid base using the built-in natural logarithm function.log_b(1) = 0 — because b⁰ = 1 for any valid baselog_b(b) = 1 — because b¹ = blog_b(xy) = log_b(x) + log_b(y) — product rulelog_b(x/y) = log_b(x) − log_b(y) — quotient rulelog_b(x^n) = n × log_b(x) — power rulePractical Examples
🇮🇳 Meera — Chemistry Student, Pune
Meera calculates the pH of a solution with hydrogen ion concentration [H⁺] = 0.001 M. pH = −log₁₀(0.001).
log₁₀(0.001) = log₁₀(10⁻³) = −3. Therefore pH = −(−3) = 3.
✓ pH = 3 — an acidic solution, verified by the definition pH = −log₁₀[H⁺].
🇮🇳 Arjun — Computer Science Student, Hyderabad
Arjun needs to find how many bits are required to represent 256 distinct values. Bits = log₂(256).
log₂(256) = log₂(2⁸) = 8.
✓ 8 bits needed — because 2⁸ = 256 distinct values. Standard byte size confirmed.
🇺🇸 Emily — Finance Analyst, New York
Emily calculates how long it takes for an investment to double at 7% annual growth. Using the rule: t = ln(2)/ln(1.07).
ln(2) ≈ 0.6931, ln(1.07) ≈ 0.0677. So t = 0.6931/0.0677 ≈ 10.24 years.
✓ Approximately 10.24 years to double — consistent with the Rule of 72 estimate (72/7 ≈ 10.3).
🇮🇳 Sunita — Physics Teacher, Jaipur
Sunita demonstrates sound intensity levels. A sound at 10⁻⁵ W/m² compared to threshold 10⁻¹² W/m²: dB = 10 × log₁₀(10⁻⁵/10⁻¹²).
10 × log₁₀(10⁷) = 10 × 7 = 70 dB.
✓ 70 decibels — equivalent to normal conversation level. Formula verified.
What Is a Logarithm Calculator?
A logarithm calculator is a mathematical tool that computes the logarithm of any positive number to any valid base. Logarithms are the inverse of exponentiation — if you know that 2⁵ = 32, then log₂(32) = 5. The logarithm tells you the exponent.
Logarithms appear everywhere in science and engineering: pH scales in chemistry, decibel measurements in acoustics, Richter scales in seismology, bit calculations in computing, and compound interest formulas in finance. Understanding them is essential across nearly every STEM field.
This calculator handles four base types — common (base 10), natural (base e ≈ 2.71828), binary (base 2), and any custom base you specify. It provides not just the answer but a complete breakdown showing the formula used, the change-of-base conversion, and antilog verification so you can understand and verify every result.
Logarithm Calculator in Multiple Languages
Want to master logarithm concepts with real-world examples?
Read Our Complete Guide to Logarithms →Frequently Asked Questions
Is this logarithm calculator free to use?
Yes, this logarithm calculator is completely free with no signup, no limits, and no data collection. All calculations run in your browser.
What is the difference between log and ln?
Log (common logarithm) uses base 10, while ln (natural logarithm) uses base e (approximately 2.71828). In mathematics, log without a base usually means base 10, but in some contexts it means natural log.
Can I calculate logarithms with any base?
Yes, this calculator supports base 10, base e (natural log), base 2, and any custom positive base that is not equal to 1.
Why can't I take the logarithm of a negative number?
In real number mathematics, logarithms are only defined for positive numbers. The logarithm of a negative number or zero is undefined because no real exponent can produce a negative result from a positive base.
What is log base 2 used for?
Log base 2 (binary logarithm) is essential in computer science for calculating bit requirements, algorithm complexity (Big O notation), and information theory measurements like entropy.
How accurate are the results?
Results use JavaScript 64-bit double precision floating-point arithmetic, providing accuracy to approximately 15-17 significant digits. Displayed values are rounded to 10 decimal places.
What is an antilogarithm?
An antilogarithm is the inverse of a logarithm. If log_b(x) = y, then the antilog is x = b^y. This calculator shows the antilog verification in the result breakdown.
Why can't the base be 1?
Base 1 is undefined for logarithms because 1 raised to any power always equals 1. There is no exponent that would produce any number other than 1, making the logarithm meaningless.
Does this tool work on mobile devices?
Yes, the logarithm calculator is fully responsive and works on smartphones, tablets, and desktops with touch-friendly controls.
Can I calculate log of very large or very small numbers?
Yes, the calculator handles numbers within JavaScript's floating-point range, from approximately 5×10⁻³²⁴ to 1.8×10³⁰⁸. Scientific notation input is supported.
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