Exponential Function Formula & Examples
1. Definition and Basic Formula
What is an Exponential Function?
An exponential function is a function in which the variable appears in the exponent. The general form is:
where \(a > 0\) and \(a \neq 1\), and \(x\) is any real number.
The Natural Exponential Function
The most important exponential function uses Euler's number \(e\) as the base:
\[f(x) = e^x\]where \(e \approx 2.71828...\)
2. General Forms of Exponential Functions
Standard Form
\[f(x) = a \cdot b^x\]- \(a\) = initial value (y-intercept)
- \(b\) = base (growth/decay factor)
- \(x\) = exponent (independent variable)
Exponential Growth Formula
When a quantity increases exponentially:
- \(a\) = initial amount
- \(r\) = growth rate (as a decimal)
- \(x\) = time
- Condition: \(r > 0\) (positive growth rate)
Exponential Decay Formula
When a quantity decreases exponentially:
- \(a\) = initial amount
- \(r\) = decay rate (as a decimal)
- \(x\) = time
- Condition: \(0 < r < 1\) (positive decay rate less than 1)
3. Exponential Function Rules
Laws of Exponents
- \(a^0 = 1\) (Zero Exponent Rule)
- \(a^{-x} = \frac{1}{a^x}\) (Negative Exponent Rule)
- \(a^x \cdot a^y = a^{x+y}\) (Product Rule)
- \(\frac{a^x}{a^y} = a^{x-y}\) (Quotient Rule)
- \((a^x)^y = a^{xy}\) (Power of a Power Rule)
- \(a^x \cdot b^x = (ab)^x\) (Product of Powers Rule)
- \(\left(\frac{a}{b}\right)^x = \frac{a^x}{b^x}\) (Quotient of Powers Rule)
4. Worked Examples
Problem: In 2010, there were 100,000 citizens in a town. If the population increases by 8% every year, then how many citizens will there be in 10 years? Round your answer to the nearest integer.
Solution:
- Initial population: \(a = 100,000\)
- Growth rate: \(r = 8\% = 0.08\)
- Time: \(x = 10\) years
Problem: Simplify the exponential expression \(5^x - 5^{x+3}\).
Solution:
Problem: Solve the equation \(4^3 \times 4^{x+5} = 4^{2x+12}\).
Solution:
Problem: You invest $5,000 at an annual interest rate of 6% compounded annually. How much will you have after 8 years?
Solution:
- \(P = \$5,000\) (principal)
- \(r = 0.06\) (6% interest rate)
- \(t = 8\) years
Problem: The half-life of a radioactive substance is 20 years. If you start with 80 grams, how much will remain after 60 years?
Solution:
- \(A_0 = 80\) grams (initial amount)
- \(t = 60\) years (time elapsed)
- \(h = 20\) years (half-life)
5. Properties of Exponential Functions
- Domain: All real numbers \((-\infty, \infty)\)
- Range: All positive real numbers \((0, \infty)\)
- Y-intercept: \((0, a)\) for \(f(x) = a \cdot b^x\)
- Horizontal Asymptote: \(y = 0\) (the x-axis)
- Increasing: If \(b > 1\), the function is increasing
- Decreasing: If \(0 < b < 1\), the function is decreasing
- One-to-one: Each x-value corresponds to exactly one y-value
6. Derivative and Integral
Derivative of Exponential Functions
For \(f(x) = e^x\):
\[\frac{d}{dx}(e^x) = e^x\]For \(f(x) = a^x\):
\[\frac{d}{dx}(a^x) = a^x \ln a\]Chain Rule for \(f(x) = e^{g(x)}\):
\[\frac{d}{dx}(e^{g(x)}) = g'(x) \cdot e^{g(x)}\]Integral of Exponential Functions
For \(f(x) = e^x\):
\[\int e^x \, dx = e^x + C\]For \(f(x) = a^x\):
\[\int a^x \, dx = \frac{a^x}{\ln a} + C\]For \(f(x) = e^{ax}\):
\[\int e^{ax} \, dx = \frac{e^{ax}}{a} + C\]7. Real-World Applications
Common Applications
- Population Growth: Modeling increases in population over time
- Compound Interest: Calculating investment growth
- Radioactive Decay: Determining half-life of substances
- Bacterial Growth: Tracking colony expansion
- Carbon Dating: Estimating age of fossils and artifacts
- Medicine: Drug concentration in bloodstream
- Physics: Cooling and heating processes (Newton's Law of Cooling)
- Economics: Inflation and depreciation calculations
Key Takeaway: Exponential functions model situations where the rate of change is proportional to the current value, leading to rapid growth or decay.