Logarithm Calculator
Calculate logarithms with any base, solve for x or base, with step-by-step solutions. Supports natural log (ln), common log (log10), and binary log (log2).
Calculate log base b of x
log
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) =
Result
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Formula Used
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High Precision
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Solve for x: log base b of x = y
log
( x ) =
x =
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Formula Used
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High Precision
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Solve for base b: log base b of x = y
log
b
(
) =
Base (b) =
-
Formula Used
-
High Precision
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Common Logarithm Types
Natural Logarithm
ln(x) = log_e(x)
Base e (2.71828...). Used in calculus, compound interest, and natural growth/decay.
Common Logarithm
log(x) = log_10(x)
Base 10. Used in pH scale, decibels, Richter scale, and scientific notation.
Binary Logarithm
lg(x) = log_2(x)
Base 2. Used in computer science, information theory, and binary systems.
Logarithm Properties
Product Rule
log_b(xy) = log_b(x) + log_b(y)
Example: log(100) = log(10) + log(10) = 1 + 1 = 2
Quotient Rule
log_b(x/y) = log_b(x) - log_b(y)
Example: log(5) = log(50) - log(10) = 1.699 - 1 = 0.699
Power Rule
log_b(x^n) = n * log_b(x)
Example: log(1000) = log(10^3) = 3 * log(10) = 3
Change of Base Formula
log_b(x) = ln(x) / ln(b)
Example: log_2(8) = ln(8) / ln(2) = 2.079 / 0.693 = 3
Log of 1
log_b(1) = 0
Any base raised to 0 equals 1, so log of 1 is always 0
Log of Base
log_b(b) = 1
Any base raised to 1 equals itself, so log_b(b) = 1