The Power of Compounding
In an Exponential Function, the variable is in the exponent. These functions describe things that grow or shrink at a rate proportional to their current size.
\[y = a \cdot b^x\]
- \(a\): Initial value (at \(x=0\)).
- \(b\): Growth factor (\(b > 1\) for growth, \(0 < b < 1\) for decay).
Worked Examples
Example 1: Population Growth
A colony of 100 bacteria doubles every hour. Find the population after 5 hours.
- \(a = 100\), \(b = 2\), \(x = 5\).
- \(y = 100 \cdot 2^5 = 100 \cdot 32 = 3200\).
- Result: 3200
Example 2: Radioactive Decay
An isotope has a half-life of 10 years. If you start with 80g, how much is left after 30 years?
- Initial \(a = 80\). Growth factor \(b = 0.5\) (half).
- Number of periods: \(30/10 = 3\).
- \(y = 80 \cdot (0.5)^3 = 80 \cdot 0.125 = 10\).
- Result: 10g
The Bridge to Quantum Mechanics
In Quantum Mechanics, particles are described by waves. When a particle encounters a wall it cannot pass through (classically), its probability doesn't drop to zero instantly. Instead, the wavefunction enters a state of Exponential Decay. The further into the wall you look, the less likely you are to find the particle. However, if the wall is thin enough, the decay doesn't reach zero before the other side, allowing the particle to "appear" on the other side. This is Quantum Tunneling, and it is how the Sun produces energy.