Exponential Calculator
Calculate exponential functions, powers, natural logarithms, and solve exponential equations with detailed explanations.
Natural Exponential Function (e^x)
Examples:
General Power Function (a^x)
Examples:
Exponential Growth/Decay
Formula: A = A₀ × e^(rt)
Examples:
Solve Exponential Equations
Examples:
Common Exponential Values
| x | e^x | ln(e^x) | 2^x | 10^x |
|---|---|---|---|---|
| 0 | 1 | 0 | 1 | 1 |
| 1 | 2.718 | 1 | 2 | 10 |
| 2 | 7.389 | 2 | 4 | 100 |
| 3 | 20.086 | 3 | 8 | 1,000 |
| -1 | 0.368 | -1 | 0.5 | 0.1 |
Exponential Functions
Natural Exponential:
f(x) = e^x
Base e ≈ 2.71828
General Exponential:
f(x) = a^x
Where a > 0, a ≠ 1
Growth/Decay:
A = A₀e^(rt)
Continuous compounding
Properties
- e^0 = 1
- e^1 = e
- e^(x+y) = e^x × e^y
- e^(x-y) = e^x / e^y
- (e^x)^y = e^(xy)
- ln(e^x) = x
- e^(ln x) = x
Applications
- Population growth
- Radioactive decay
- Compound interest
- Cooling/heating
- Pharmacokinetics
- Electronics
Understanding Exponential Functions
Exponential functions are mathematical functions of the form f(x) = a^x, where the base 'a' is a positive constant and the variable 'x' is the exponent. The most important exponential function uses Euler's number e ≈ 2.71828 as the base.
Key Characteristics:
- Domain: All real numbers
- Range: (0, ∞) for a > 1
- Passes through (0, 1)
- Continuous and smooth
- One-to-one function
Growth vs Decay
Exponential Growth (r > 0):
Function increases rapidly as x increases
Examples: Population growth, compound interest
Exponential Decay (r < 0):
Function decreases rapidly toward zero
Examples: Radioactive decay, cooling
Half-life Formula:
A(t) = A₀ × (1/2)^(t/h), where h is half-life