Calculate an, ex, 10x & Exponential Growth/Decay | Step-by-step Solutions
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📝 Step-by-step solution
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💡 an = a raised to power n | e ≈ 2.71828 | exp(x) = ex | Growth: A = P × (1 + r)t
📖 What is an Exponential Function?
An exponential function is a mathematical function of the form f(x) = ax,
where a is a positive constant (the base) and x is the exponent.
Exponential functions describe rapid growth or decay and appear everywhere — from compound interest and population growth
to radioactive decay and viral spread.
The most important exponential base is e ≈ 2.71828 (Euler's number). The natural exponential function
ex appears in calculus, physics, biology, and finance.
🧮 Key Exponential Formulas
General power: an = a × a × ... × a (n times) Natural exponential: exp(x) = ex Power of 10: 10x (scientific notation base) Exponential growth: A = P × (1 + r)t Exponential decay: A = P × (1 − r)t Continuous growth: A = P × ert
📝 Worked Examples
Example 1: General Power — 210
Given: a = 2, n = 10 Formula: an = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 Result:1024
Example 2: Natural Exponential — e2
Given: x = 2, e ≈ 2.71828 Formula: e2 = 2.71828 × 2.71828 Result:≈ 7.38906
Example 3: Exponential Growth — $1,000 at 8% for 10 years
Given: P = 1000, r = 8% = 0.08, t = 10 years Formula: A = 1000 × (1 + 0.08)10 Result: A = 1000 × 2.15892 = $2,158.92
📊 Properties of Exponents
Product rule: am × an = am+n
Quotient rule: am ÷ an = am−n
Power rule: (am)n = am×n
Zero exponent: a0 = 1 (a ≠ 0)
Negative exponent: a−n = 1 / an
Fractional exponent: a1/n = ⁿ√a
💡 Real-World Applications
Compound Interest: Money grows exponentially over time
Population Growth: Species reproduce at exponential rates
an is a general exponential with any positive base a.
ex is a special case where the base is Euler's number e ≈ 2.71828.
The natural exponential has unique calculus properties (derivative of ex is ex).
What is exponential growth?
Exponential growth means a quantity increases by a constant percentage per unit time.
For example, 10% annual growth means each year the value is 1.10× the previous year.
The formula is A = P × (1 + r)t.
What is exponential decay?
Exponential decay means a quantity decreases by a constant percentage per unit time.
The formula is A = P × (1 − r)t. Radioactive decay is the classic example.
What is the Rule of 72?
The Rule of 72 estimates how long it takes for an exponentially growing quantity to double:
72 ÷ growth rate (%) = years to double. At 8% growth, doubling takes ~9 years.
Why is e used in finance?
Continuous compounding uses e: A = P × ert. This represents the theoretical limit of
compounding infinitely often, giving the highest possible growth.
This Exponential Calculator is provided for informational and educational purposes only.
Results are based on standard mathematical formulas and are rounded for readability.
For critical scientific, engineering, or financial applications, always verify with authoritative sources
or a qualified professional. Online Calculator Zone is not responsible for any decisions made based on
these calculations.