Calculate logarithms with any base: log₁₀, ln (natural log), log₂, or custom base. Includes antilog calculator and logarithm rules reference.
log_10
2
Common logarithm calculations
Verification:
102.000000 = 100
Antilog is the inverse of log: if log_b(x) = y, then antilog_b(y) = x = b^y
Interactive formula breakdown
The logarithm operation - finds the exponent
The base of the logarithm
The number we are taking the logarithm of
The exponent: b^y = x
Note: log(1) = 0 for all bases, log crosses x-axis at x = 1
Product Rule
log(ab) = log(a) + log(b)
Quotient Rule
log(a/b) = log(a) - log(b)
Power Rule
log(aⁿ) = n · log(a)
Zero Rule
log(1) = 0
Identity Rule
log_b(b) = 1
Change of Base
log_b(x) = log(x)/log(b)
e (Euler's Number)
2.71828182845904...
Base of natural logarithms
log₁₀(e)
0.43429448190325...
Common log of e
ln(10)
2.30258509299404...
Natural log of 10
log₂(10)
3.32192809488736...
Binary log of 10
Test your understanding with real problems
Practice with 8 problems to test your understanding.
log_10
2
A logarithm answers 'what power gives this value?' If log_b(x) = y, then b^y = x. Common logs: log10(1000) = 3 (10^3 = 1000), ln(e) = 1 (e^1 = e). Key rules: log(ab) = log(a) + log(b), log(a/b) = log(a) - log(b), log(a^n) = n*log(a). Change of base: log_b(x) = log(x)/log(b).
A logarithm is the inverse operation of exponentiation. The logarithm base b of a number x (written log_b(x)) is the exponent to which b must be raised to produce x. Logarithms simplify multiplication to addition, making them historically crucial for calculations and still fundamental in mathematics, science, and engineering.
A logarithm answers 'what power gives this value?' If log_b(x) = y, then b^y = x. Common logs: log10(1000) = 3 (10^3 = 1000), ln(e) = 1 (e^1 = e). Key rules: log(ab) = log(a) + log(b), log(a/b) = log(a) - log(b), log(a^n) = n*log(a). Change of base: log_b(x) = log(x)/log(b).
A logarithm answers the question: "To what power must the base be raised to get this number?" For example, log₂(8) = 3 because 2³ = 8. Logarithms are the inverse of exponential functions.
log (common logarithm) uses base 10 and is used in science and engineering. ln (natural logarithm) uses base e ≈ 2.718 and is used in calculus and natural growth. log₂ (binary logarithm) uses base 2 and is common in computer science.
Logarithms are undefined for zero and negative numbers. log(0) is undefined because no power of a positive base equals 0. log of negative numbers is undefined in real numbers (complex in advanced math). The base must be positive and not equal to 1.
Key rules: log(ab) = log(a) + log(b) (product), log(a/b) = log(a) - log(b) (quotient), log(aⁿ) = n·log(a) (power). Also: log(1) = 0, log(base) = 1, and change of base: log_b(x) = log(x)/log(b).
Use the change of base formula: log_b(x) = log(x) / log(b) = ln(x) / ln(b). Any common logarithm (base 10) or natural logarithm can be converted to any base using this formula.
e ≈ 2.71828 is the base of natural logarithms. It's a mathematical constant that appears in compound interest, population growth, and calculus. The function eˣ is unique because its derivative equals itself.
Last updated: 2025-01-15
log_10
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