Exponents & Log Calculator

Calculate Exponents & Logarithms

Choose a mode and enter your mathematical values

Choose exponent or log mode, enter numbers, then press Calculate.

Base (a):

Can be any real number (positive, negative, or zero)

Exponent (n):

Can be integer, fraction, or decimal (positive or negative)

Results & Actions

Calculate and view your mathematical results

Mathematical Expression

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Numerical Result

Scientific Notation

Calculation Type

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How It Works

Select exponent or logarithm mode, enter your values, and get instant results. Our calculator uses precise mathematical formulas for accuracy in academic and professional applications.

Common Uses

Calculate compound interest, scientific notation, pH values, earthquake magnitudes, sound decibels, and exponential growth/decay with this versatile mathematical tool.

Always Accessible

Works completely in your browser - no data sent to servers. Use it anytime, anywhere with full privacy protection for homework, research, or professional calculations.

Mathematical Formulas

Basic Exponent Rule

aⁿ = a × a × a × ... (n times)

a = base number, n = exponent (positive integer)

Negative Exponent Rule

a⁻ⁿ = 1 / aⁿ

a = base number, n = positive exponent, result is reciprocal

Fractional Exponent Rule

a^(m/n) = ⁿ√(aᵐ) = (ⁿ√a)ᵐ

a = base, m/n = fraction, represents roots and powers

Logarithm Definition

If aˣ = b, then logₐ(b) = x

a = base, b = value, x = exponent that produces b

Change of Base Formula

logₐ(b) = logₓ(b) / logₓ(a)

a = original base, b = value, x = any positive base ≠ 1

Power Rule for Logarithms

logₐ(xⁿ) = n × logₐ(x)

a = base, x = value, n = exponent

Step-by-Step Examples

Compound Interest (Exponent)

$1,000 at 5% for 10 years: 1000 × (1.05)^10

Future value: $1,628.89

Earthquake Magnitude (Logarithm)

Compare 5.0 vs 7.0 Richter: 10^(1.5 × 2)

Energy ratio: 1,000 times greater

Sound Intensity (Combined)

0.001 W/m² to decibels: 10 × log₁₀(10⁹)

Sound level: 90 dB

Understanding Exponents & Logarithms

Exponents and logarithms are fundamental mathematical operations used across science, engineering, finance, and data analysis. Understanding these concepts helps solve problems involving growth, scaling, and measurement across different orders of magnitude.

What are Exponents?

Exponents represent repeated multiplication. The expression aⁿ means multiply a by itself n times. Exponents are essential for calculating compound interest, population growth, radioactive decay, and scientific notation for very large or small numbers.

What are Logarithms?

Logarithms are the inverse operation of exponents. If aˣ = b, then logₐ(b) = x. Logarithms compress large ranges into manageable scales, making them ideal for comparing quantities that span multiple orders of magnitude, like earthquake energies or sound intensities.

Real-World Applications

Finance: Compound interest calculations use exponents to determine investment growth over time
Chemistry: pH scale uses base-10 logarithms to measure acidity: pH = -log₁₀[H⁺]
Geology: Richter scale uses base-10 logarithms for earthquake magnitude comparisons
Acoustics: Decibel scale uses base-10 logarithms for sound intensity measurements
Computer Science: Algorithm complexity (Big O notation) often involves logarithmic terms

Practical Example: Compound Interest Calculation

Imagine investing $1,000 at 5% annual interest compounded annually for 10 years.

Using Exponent Calculation

Formula: A = P(1 + r)^t

P = $1,000 (principal)
r = 0.05 (5% interest rate)
t = 10 years

Calculation: 1.05^10 = 1.628894626777

Result: $1,000 × 1.62889 = $1,628.89

Using Logarithm to Find Time

If you want to know how long it takes to double your money at 5% interest:

Formula: t = log(2) / log(1.05)

Calculation: log₁₀(2) ≈ 0.3010, log₁₀(1.05) ≈ 0.02119

Result: 0.3010 ÷ 0.02119 ≈ 14.2 years

Frequently Asked Questions

What's the difference between exponents and logarithms?

Exponents answer "what is a raised to the power n?" while logarithms answer "what power must we raise a to get b?" They're inverse mathematical operations.

When should I use natural log (ln) versus common log (log₁₀)?

Use natural log for continuous growth models, calculus, and natural sciences. Use common log for engineering, pH calculations, and decimal-based measurements.

Why can't I use negative numbers for logarithms?

Logarithms of negative numbers are undefined in real numbers because no real exponent of a positive base yields a negative result. For negative values, you'd need complex number calculations.

How do fractional exponents work in real-world terms?

Fractional exponents represent roots. For example, 4^(1/2) = √4 = 2, and 8^(2/3) = (∛8)² = 2² = 4. They're useful in geometry, physics, and engineering.

What does a negative exponent mean practically?

Negative exponents represent reciprocals. For example, 2⁻³ = 1/2³ = 1/8. This is useful in science for units (like cm⁻¹) and in finance for present values.

How accurate are the calculator's results?

The calculator uses double-precision floating-point arithmetic, providing about 15-17 significant digits of accuracy. For most academic and professional applications, this exceeds required precision.