Exponents & Logarithms Calculator
Calculate powers, roots, common logs, natural logs, and solve exponential equations
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⚡ Exponents
Repeated multiplication. Example: 2³ = 8. Used for compound interest, scientific notation, population growth, and computer science.
📊 Logarithms
The inverse of exponents. log₁₀(1000) = 3 because 10³ = 1000. Used for pH scale, earthquake magnitudes, and decibels.
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📐 Key Formulas & Rules
Basic Exponent Rule
Example: 2⁴ = 2 × 2 × 2 × 2 = 16
Negative Exponent
Example: 2⁻³ = 1/8 = 0.125
Fractional Exponent (Roots)
Example: 8^(1/3) = ³√8 = 2
Logarithm Definition
Example: log₁₀(100) = 2 because 10² = 100
Change of Base Formula
Use any base (common log or natural log) to calculate custom logs
🎯 Real-World Examples
Invest $1,000 at 5% for 10 years: 1000 × (1.05)^10 = $1,628.89
A magnitude 7 earthquake is 10^(1.5×2) = 1000× more energetic than magnitude 5.
Intensity 0.001 W/m² → 10 × log₁₀(10⁹) = 90 dB
❓ Frequently Asked Questions
Exponents ask "a to power n equals what?" Logs ask "what power gives this result?" They're inverses.
Use ln for calculus, growth models, and natural sciences. Use log₁₀ for engineering, pH, decibels, and Richter scale.
Logs of negative numbers are not real — no real exponent of a positive base gives a negative result.
a^(1/n) means the nth root of a. Example: 16^(1/4) = ⁴√16 = 2.
It means reciprocal. Example: 10⁻³ = 1/1000 = 0.001.
