Logarithm Calculator
Free logarithm calculator. Calculate log base 10, natural log (ln), log base 2, and any custom base logarithm with step-by-step explanation.
Logarithm Calculator
Comprehensive Guide: Logarithms, Change of Base & Exponential Inverses
Master natural logs (ln), common logs (log10), algebraic logarithmic identities, the Richter scale, pH chemistry, and half-life decay.
1. Mathematical Definition of Logarithms
A logarithm is the inverse mathematical operation to exponentiation. It answers the fundamental algebraic question: "To what power must base b be raised to produce number x?"
Logarithms cannot accept zero or negative arguments in the real number system because raising a positive base to any real exponent can never yield a negative number or zero.
2. Core Logarithmic Identities & The Change of Base Formula
| Property Name | Algebraic Identity | Computational Purpose |
|---|---|---|
| Product Rule | log_b(xy) = log_b(x) + log_b(y) | Converts multiplication into simple addition |
| Quotient Rule | log_b(x / y) = log_b(x) − log_b(y) | Converts division into subtraction |
| Power Rule | log_b(xᵏ) = k · log_b(x) | Brings exponents down into linear multipliers |
| Change of Base | log_b(x) = ln(x) / ln(b) = log₁₀(x) / log₁₀(b) | Allows computing any base on standard calculators |
3. Graduated Step-by-Step Worked Problems
Problem 1: Solving an Exponential Equation with Logarithms
Solve for x: 5^(2x − 1) = 1250
Step 1: Take the natural logarithm (ln) of both sides:
ln(5^(2x − 1)) = ln(1250).
Step 2: Bring exponent down using the power rule:
(2x − 1) · ln(5) = ln(1250).
Step 3: Divide by ln(5):
2x − 1 = ln(1250) / ln(5) ≈ 7.130899 / 1.609438 ≈ 4.4307.
Step 4: Solve for x:
2x = 4.4307 + 1 = 5.4307 ⟹ x ≈ 2.7153.
Result: x ≈ 2.715.
Problem 2: Evaluating Arbitrary Base via Change of Base
Evaluate: log₂(256) and log₅(100)
• log₂(256) = ln(256) / ln(2) = 5.545177 / 0.693147 = 8.0 (since 2⁸ = 256).
• log₅(100) = log₁₀(100) / log₁₀(5) = 2.0 / 0.69897 ≈ 2.8614.
Result: log₂(256) = 8; log₅(100) ≈ 2.861.
Problem 3: Decibel Sound Intensity Ratio
How many times more acoustically powerful is a rock concert at 110 dB than conversational speech at 60 dB?
• Formula: ΔdB = 10 · log₁₀(I₂ / I₁).
• 110 − 60 = 50 dB = 10 · log₁₀(Ratio) ⟹ log₁₀(Ratio) = 5.
• Ratio = 10⁵ = 100,000.
Result: The concert is 100,000 times more powerful.
4. Logarithmic Scales in Science: pH, Decibels & Richter Scale
Logarithms compress enormous dynamic ranges into manageable scales:
- pH Acidity Scale (Chemistry):
pH = −log₁₀[H⁺]. A drop of 1 pH unit represents a 10-fold increase in hydrogen ion concentration! - Decibels dB (Acoustics):
dB = 10 · log₁₀(I / I₀). An 80 dB sound possesses 100 times the acoustic power of a 60 dB sound. - Richter Magnitude Scale (Seismology): Each integer step represents a
10^(1.5) ≈ 31.6times increase in released seismic energy.
5. Frequently Asked Questions (FAQ)
What is the difference between log and ln?
log(x) (common log) defaults to base 10. ln(x) (natural log) is strictly base e ≈ 2.71828. In computer science and programming languages (like Python, C, JavaScript), Math.log(x) evaluates natural log base e.
Can you take the logarithm of a negative number?
In real number arithmetic, logarithms of negative numbers are undefined. In complex analysis, Euler's formula e^(iπ) = −1 yields complex logarithms: ln(−x) = ln(x) + iπ.
Why were logarithms historically so important before electronic computers?
John Napier invented logarithms in 1614 to transform tedious multi-digit multiplication and division into simple addition and subtraction. For 350 years, astronomers, navigators, and engineers used printed book tables of logarithms and slide rules for all major calculations.