Logarithm Calculator
Calculate logarithms with any base, including natural log and common log
Logarithm Formulas
What is a Logarithm?
A logarithm answers the question: 'To what power must we raise the base to get this number?' If log₂(8) = 3, it means 2³ = 8. Logarithms are the inverse operation of exponentiation.
There are two commonly used logarithm bases: natural logarithm (ln) with base e ≈ 2.718, used extensively in calculus and natural sciences; and common logarithm (log₁₀) with base 10, used in engineering and the decibel scale.
Logarithms transform multiplication into addition, which historically made complex calculations possible before computers. Today, they're essential in measuring earthquake intensity (Richter scale), sound levels (decibels), pH in chemistry, and exponential growth/decay.
Types of Logarithms
Common Log (log₁₀)
Base 10. Used for decibels, pH scale, Richter scale. log₁₀(100) = 2.
Natural Log (ln)
Base e ≈ 2.718. Used in calculus, continuous growth, physics. ln(e) = 1.
Binary Log (log₂)
Base 2. Used in computer science for bit calculations. log₂(8) = 3.
Custom Base
Any positive base except 1. Converted using change of base formula.
Logarithm Laws
Master these rules to simplify logarithmic expressions:
| Rule | Formula | Example | Result |
|---|---|---|---|
| Product | log(xy) = log(x) + log(y) | log(2×5) | log(10) = 1 |
| Quotient | log(x/y) = log(x) - log(y) | log(100/10) | 2 - 1 = 1 |
| Power | log(xⁿ) = n·log(x) | log(10²) | 2·log(10) = 2 |
| Root | log(√x) = log(x)/2 | log(√100) | 2/2 = 1 |
| Identity | log_b(b) = 1 | log₁₀(10) | = 1 |
| Zero | log_b(1) = 0 | log₁₀(1) | = 0 |
Real-World Applications
Richter Scale
Earthquake magnitude is log₁₀ of amplitude. Each whole number increase = 10× more ground motion. A magnitude 6 is 10× stronger than magnitude 5.
Decibels
Sound intensity measured as dB = 10·log₁₀(I/I₀). Every 10 dB increase = 10× more sound intensity. 90 dB is 10× louder than 80 dB.
pH Scale
pH = -log₁₀[H⁺]. Each pH unit = 10× difference in acidity. pH 4 is 10× more acidic than pH 5.
Exponential Growth
Logarithms reveal doubling time in growth. Time to double = ln(2)/growth rate ≈ 0.693/r.
Frequently Asked Questions
Why can't I take the log of a negative number?
In real numbers, no power of a positive base produces a negative result. log(-8) is undefined in real numbers. Complex logarithms extend to negative numbers using imaginary numbers.
What's the difference between log and ln?
log typically means log₁₀ (common log), while ln is logₑ (natural log, base e). In pure mathematics, 'log' often means natural log, but in engineering and calculators, it usually means log₁₀.
How do I solve logarithmic equations?
Convert to exponential form: if log₂(x) = 5, then x = 2⁵ = 32. For equations like log(x) + log(x-3) = 1, combine using product rule, then convert to exponential form.
What is e and why is it important?
e ≈ 2.71828 is Euler's number, the base of natural logarithms. It appears naturally in compound interest, growth/decay, and calculus. It's defined as lim(1 + 1/n)ⁿ as n→∞.
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