Exponential Function Calculator
Calculate exponential functions of the form f(x) = a · bcx
Common bases: e ≈ 2.718 (natural), 10 (common), or any positive number ≠ 1
This multiplies x in the exponent: bcx
How to Use This Calculator
Enter Coefficient
Input the coefficient (a) that multiplies the exponential term. Default is 1. For example, if f(x) = 3 × 2^x, enter 3.
Enter Base
Input the base (b) of the exponential function. Must be positive and not equal to 1. Common choices: e ≈ 2.718, 10, or 2.
Enter Exponent Coefficient
Input the coefficient (c) that multiplies x in the exponent. For f(x) = e^(2x), enter 2. For f(x) = 10^x, enter 1.
Enter x Value
Input the value of x at which you want to evaluate the function. This can be any real number.
Calculate
Click "Calculate f(x)" to compute the value of the exponential function at the given x.
Formula
f(x) = a · bcx
Where:
- a = coefficient (constant multiplier)
- b = base (positive number, not equal to 1)
- c = exponent coefficient (multiplies x in the exponent)
- x = variable (input value)
Example 1: f(x) = 2x, find f(3)
f(3) = 2³ = 8
Here: a = 1, b = 2, c = 1, x = 3
Example 2: f(x) = 3 × e2x, find f(1)
f(1) = 3 × e2×1 = 3 × e² ≈ 3 × 7.389 ≈ 22.167
Here: a = 3, b = e, c = 2, x = 1
Example 3: f(x) = 5 × 10-0.5x, find f(2)
f(2) = 5 × 10-0.5×2 = 5 × 10⁻¹ = 5 × 0.1 = 0.5
Here: a = 5, b = 10, c = -0.5, x = 2
About Exponential Function Calculator
The Exponential Function Calculator evaluates exponential functions of the form f(x) = a · bcx. Exponential functions are fundamental in mathematics, modeling exponential growth and decay, compound interest, radioactive decay, population growth, and many other natural phenomena. This calculator helps you quickly evaluate exponential functions at specific values of x.
When to Use This Calculator
- Compound Interest: Calculate investment growth: A = P(1 + r)t or continuous compounding: A = Pert
- Population Growth: Model population growth: P(t) = P₀ × ert
- Radioactive Decay: Calculate remaining material: N(t) = N₀ × e-λt
- Exponential Growth: Model bacterial growth, virus spread, or financial investments
- Calculus: Evaluate exponential functions before differentiating or integrating
Why Use Our Calculator?
- ✅ Instant Results: Get accurate function values immediately
- ✅ Flexible: Supports any base, coefficient, and exponent
- ✅ Clear Formula: Shows the formula being used in the calculation
- ✅ Educational: Learn exponential functions through examples
- ✅ 100% Free: No registration or payment required
- ✅ High Precision: Accurate calculations with scientific notation for large results
Common Applications
Finance - Compound Interest: If you invest $1,000 at 5% annual interest, compounded continuously for 10 years: A = 1000 × e0.05×10 = 1000 × e0.5 ≈ $1,648.72.
Biology - Population Growth: If a bacterial population doubles every 2 hours, after 6 hours: P(6) = P₀ × 26/2 = P₀ × 2³ = P₀ × 8 (8 times the initial population).
Physics - Radioactive Decay: If a radioactive substance has a half-life of 5 years, after 15 years: N(15) = N₀ × (1/2)15/5 = N₀ × (1/2)³ = N₀/8 (1/8 of the original amount).
Tips for Best Results
- The base (b) must be positive and not equal to 1
- For natural exponential functions, use base e ≈ 2.71828
- For exponential growth, the exponent coefficient (c) is positive
- For exponential decay, the exponent coefficient (c) is negative
- When c = 0, the function becomes constant: f(x) = a
- Large positive exponents give very large results
- Large negative exponents give values close to zero
Frequently Asked Questions
What is an exponential function?
An exponential function is a function of the form f(x) = a · bcx, where b is a positive constant (not equal to 1), a is a coefficient, and c multiplies x in the exponent. The variable x appears in the exponent.
What's the difference between exponential and polynomial functions?
In exponential functions (like 2x), the variable is in the exponent. In polynomial functions (like x²), the variable is in the base and has fixed exponents. Exponential functions grow much faster than polynomials.
What does the coefficient (a) do?
The coefficient (a) is a constant multiplier that scales the entire function. For example, 3 × 2x is three times larger than 2x at every point.
What happens when the exponent coefficient is negative?
When c < 0, the function represents exponential decay. As x increases, f(x) decreases. For example, e-x decreases as x increases.
Why can't the base be 1?
If the base is 1, then 1cx = 1 for all x, making it a constant function rather than an exponential function. That's why b ≠ 1 is required.