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

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Solve for x: log base b of x = y

log
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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 (
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Formula Used
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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

Frequently Asked Questions

What is a logarithm and how does it work?

A logarithm answers the question: to what power must the base be raised to get a certain number? For example, log base 2 of 8 equals 3 because 2^3 = 8. The logarithm is the inverse operation of exponentiation.

What is the difference between natural log (ln) and common log (log)?

Natural log (ln) uses base e (approximately 2.71828) and is common in calculus and natural sciences. Common log (log or log10) uses base 10 and is often used in engineering and measuring sound (decibels) or earthquake intensity.

How do you calculate a logarithm with a different base?

Use the change of base formula: log_b(x) = ln(x) / ln(b) or log_b(x) = log10(x) / log10(b). This allows you to calculate any logarithm using natural log or common log functions available on calculators.