Logarithm Calculator
Calculate logarithms with any base instantly. Supports log10, ln, and log2. Includes graph and formulas.
Formula
y = log_b(x)
Finds the exponent y such that b^y = x.
Worked Examples
Example 1: Common, Binary and Natural Logs
Problem:Evaluate log10(100), log2(8) and ln(e).
Solution:A logarithm answers the question "what exponent turns the base into this number?" log10(100): 10^2 = 100, so the answer is 2 log2(8): 2^3 = 8, so the answer is 3 ln(e): e^1 = e, so the answer is 1 Note that log10(1) = log2(1) = ln(1) = 0 for the same reason: any base to the power 0 is 1.
Result:log10(100) = 2, log2(8) = 3, ln(e) = 1
Example 2: Change of Base for an Awkward Base
Problem:Evaluate log5(200), a base most calculators do not have a key for.
Solution:Change of base: log_b(x) = ln(x) / ln(b) (natural logs work, and so do base-10 logs). ln(200) = 5.298317 ln(5) = 1.609438 log5(200) = 5.298317 / 1.609438 = 3.29203 Check: 5^3.29203 = 200.0, so the exponent is right. Sanity check without a calculator: 5^3 = 125 and 5^4 = 625, so the answer must sit between 3 and 4, closer to 3.
Result:log5(200) = 3.29203
Example 3: Log Laws Turn Multiplication into Addition
Problem:Given log10(2) = 0.30103 and log10(3) = 0.47712, find log10(6), log10(1.5) and log10(8).
Solution:Product rule: log(xy) = log(x) + log(y) log10(6) = log10(2) + log10(3) = 0.30103 + 0.47712 = 0.77815 Quotient rule: log(x/y) = log(x) - log(y) log10(1.5) = log10(3) - log10(2) = 0.47712 - 0.30103 = 0.17609 Power rule: log(x^n) = n log(x) log10(8) = log10(2^3) = 3 × 0.30103 = 0.90309 These three rules are exactly what made slide rules and log tables work for three centuries.
Result:log10(6) = 0.77815, log10(1.5) = 0.17609, log10(8) = 0.90309
Frequently Asked Questions
What is a logarithm?
A logarithm is the inverse operation to exponentiation. It answers the question: "To what power must I raise the base b to get x?"
What is a natural log (ln)?
The natural logarithm (ln) has the base "e" (Euler's number, ≈ 2.718). It is fundamental in calculus and continuous growth models.
What is a common log?
The common log has base 10. It is typically written as log(x) without a subscript. It is used in engineering and the Richter scale.
Can you take the log of a negative number?
No. In the real number system, you cannot take the logarithm of a negative number or zero. The domain of log(x) is x > 0.
What is log(1)?
log(1) = 0 for any base, because any non-zero number raised to the 0th power is 1.
What is the Power Rule?
log(a^n) = n × log(a). The exponent comes out front as a multiplier.
What is the Change of Base Formula?
log_b(x) = log_k(x) / log_k(b). This allows you to calculate logs of any base using only ln or log10 button on a calculator.
Who invented logarithms?
John Napier introduced logarithms in the early 17th century to simplify complex calculations in astronomy and navigation.
Background & Theory
Logarithm:
The inverse function to exponentiation.
log_b(x) = y ⟺ b^y = x
Common Bases:
- Common Log: Base 10 (log x). Used in science/engineering.
- Natural Log: Base e (ln x). Used in math/physics. e ≈ 2.718.
- Binary Log: Base 2 (lg x). Used in computer science.
Properties:
- log(ab) = log(a) + log(b)
- log(a/b) = log(a) - log(b)
- log(a^n) = n·log(a)