Math · Logarithms
Logarithm
Calculator
Solve log_b(x)=y for any unknown. Calculate log base 10, natural log (ln), log base 2, or any custom base. Includes antilog, change of base, and full log rules reference.
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log_b(x)
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ln(x)
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log₁₀(x)
Solve log_b(x) = y
log10
(100)
=
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base
x
y
💡 Leave the field you want to solve empty. Formula: b^y = x | log_b(x) = ln(x)/ln(b)
Result
log₁₀(100)
2
dimensionless
Antilog (b^y)
100
Check b^y=x
✓
log_b(x)
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log₁₀(x)
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ln(x)
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log₂(x)
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📊 All Log Bases — Same x Value
🔄 Antilogarithm (Inverse Log)
📈 Logarithm Curves — log₁₀, ln, log₂
📐 Logarithm Rules Reference
Product Rule
log_b(xy) = log_b(x) + log_b(y)
log(2×50) = log2 + log50 = log100 = 2
Quotient Rule
log_b(x/y) = log_b(x) − log_b(y)
log(1000/10) = log1000 − log10 = 3−1 = 2
Power Rule
log_b(xⁿ) = n × log_b(x)
log(10³) = 3 × log10 = 3 × 1 = 3
Change of Base
log_b(x) = log(x) / log(b) = ln(x) / ln(b)
log₂(8) = log(8)/log(2) = 3
Identity
log_b(b) = 1 and log_b(1) = 0
log₁₀(10)=1, log₁₀(1)=0
Inverse
b^(log_b(x)) = x and log_b(b^x) = x
10^(log100) = 100
Reciprocal Base
log_b(x) = 1 / log_x(b)
log₂(8) = 1/log₈(2) = 3
Natural Log
ln(x) = log(x) / log(e) ≈ log(x) / 0.4343
ln(100) = log(100)/0.4343 ≈ 4.6052
Understanding Logarithms
A logarithm is the inverse of exponentiation. If b^y = x, then log_b(x) = y. In other words: "to what power must we raise b to get x?" Logarithms are fundamental to algebra, calculus, information theory, signal processing, and many scientific fields.
Key Logarithm Formulas
Definition: log_b(x) = y ↔ b^y = x
Common log: log(x) = log₁₀(x)
Natural log: ln(x) = log_e(x) where e ≈ 2.71828
Binary log: lb(x) = log₂(x)
Change of base: log_b(x) = ln(x)/ln(b) = log(x)/log(b)
Find x from y: x = b^y (antilogarithm)
Find base: b = x^(1/y) = e^(ln(x)/y)
Where Logarithms Are Used
Logarithms appear in pH (acid/base chemistry), decibel scale (sound), Richter scale (earthquakes), information entropy (computer science), compound interest (finance), Moore's law, and Big-O algorithm analysis. The natural log is the basis of calculus (derivative of ln(x) = 1/x).
Why can't you take the log of a negative number?
Logarithms are defined only for positive real numbers because no real power of a positive base equals a negative number or zero. For example, 10^y is always positive for any real y. The domain of log_b(x) is x > 0 and b > 0, b ≠ 1. In complex mathematics, logarithms of negative numbers do exist but produce complex values.
What is the natural logarithm and why is it "natural"?
The natural logarithm ln(x) uses the base e ≈ 2.71828. It is "natural" because it arises spontaneously in calculus: the integral of 1/x is ln(x), and e^x is the only function that is its own derivative. Ln(x) also describes natural growth and decay processes — population growth, radioactive decay, compound interest continuously compounded — making it the most mathematically fundamental logarithm.
How do logarithms relate to the pH scale?
pH = −log₁₀[H⁺] where [H⁺] is hydrogen ion concentration in moles/liter. Because the scale is logarithmic, each pH unit represents a 10× change in acidity. pH 4 is 10 times more acidic than pH 5, and 100 times more acidic than pH 6. Pure water has pH 7 (neutral), strong acids pH 1–2, strong bases pH 12–14.
What is log₂ and where is it used?
Log base 2 (binary logarithm) measures information content — log₂(x) tells you how many bits are needed to represent x possibilities. It is the foundation of information theory and computer science: a 64-bit number can represent 2^64 values, and log₂(2^64) = 64. It also appears in algorithm analysis: binary search on n elements takes log₂(n) steps.