Log Calculator
Calculate log₁₀(x) and antilog, with formula and step-by-step explanation.
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What Is a Logarithm?
A logarithm answers the question: "To what power must I raise the base to get this number?" log₁₀(1000) = 3 because 10³ = 1000. Logarithms are the inverse of exponentiation — just as subtraction undoes addition, logarithms undo exponents.
Common Logarithm (Base 10)
The common logarithm, written log(x) or log₁₀(x), uses base 10. It measures orders of magnitude: log(10) = 1, log(100) = 2, log(1000) = 3. Each increase of 1 in the log corresponds to a 10× increase in the number. Used in the Richter scale (earthquake magnitude), decibels (sound intensity), and pH (acidity).
Antilogarithm
The antilog of y (base 10) is 10^y. If log(x) = y, then x = 10^y. The antilog is used to reverse a logarithm calculation.
Logarithm Laws
log(a × b) = log(a) + log(b), log(a/b) = log(a) − log(b), log(aⁿ) = n × log(a). These properties let you turn multiplication/division into addition/subtraction, which historically enabled slide rules to perform multiplication.
Change of Base Formula
log_b(x) = log(x) / log(b) = ln(x) / ln(b). This converts any logarithm to base 10 or natural log for calculation.
Worked Example: Earthquake Magnitudes
The Richter scale uses base-10 logarithms. An earthquake measuring 7.0 is 10× more powerful than a 6.0, and 100× more powerful than a 5.0. To find by how much a magnitude 7.5 quake exceeds a 6.0: 10^(7.5−6.0) = 10^1.5 ≈ 31.6 times more energy.
Using the change-of-base formula: log₂(1000) = log(1000)/log(2) = 3/0.30103 ≈ 9.97. This tells us 1000 ≈ 2^9.97 — just under 2¹⁰ = 1024, which is why 10 binary digits can represent up to 1,023.
Common Logarithm Values Reference
| x | log₁₀(x) | Application |
|---|---|---|
| 1 | 0 | Neutral reference point |
| 2 | 0.3010 | Doubling ≈ +0.3 on log scale |
| 10 | 1 | One order of magnitude |
| 100 | 2 | pH of battery acid (~0.5) |
| 1,000 | 3 | Richter scale (3.0 quake) |
| 1,000,000 | 6 | Magnitude 6.0 earthquake |
Frequently Asked Questions
log₁₀(1000) = 3, because 10³ = 1000.
log(1) = 0 for any base, because any number to the power 0 is 1.
log usually means log₁₀ (common logarithm, base 10). ln is the natural logarithm, base e ≈ 2.71828. Both are logarithms — they just use different bases.
Not in real numbers. Logarithms are only defined for positive real numbers. log(0) is undefined (approaches −∞), and log of a negative number is complex.
The Richter scale for earthquakes, decibel scale for sound, pH scale for acidity, and many financial and scientific formulas all use base-10 logarithms.
Sound intensity in decibels (dB) = 10 × log₁₀(I/I₀), where I₀ is the reference intensity (threshold of hearing). A sound at 10× the reference is 10 dB; 100× is 20 dB; 1,000,000× is 60 dB (normal conversation). The logarithmic scale compresses the enormous range of human hearing (about 10¹²) into a convenient 0–120 dB scale.
Log base 2 (binary logarithm) is used extensively in computer science. It tells you how many binary digits (bits) are needed to represent a number: log₂(256) = 8 bits. It also measures information in bits, appears in time complexity of algorithms (binary search is O(log₂ n)), and is fundamental to data compression.