Math
Master the quantitative skills used in science and healthcare: significant figures, unit conversions, proportions, functions, exponents, logarithms, statistics, and introductory calculus.
Exponential Equations & Models
Sign in to track progressExponential models describe quantities that grow or decay by a constant percentage over time. They are widely used in population biology, radioactive decay, and compound interest.
Exponential Growth Growth occurs when the base is greater than 1:
P(t) = P₀ · (1 + r)ᵗ
where P₀ is the initial amount, r is the growth rate, and t is time.
Exponential Decay Decay occurs when the base is between 0 and 1:
P(t) = P₀ · (1 - r)ᵗ
or equivalently P(t) = P₀ · e⁻ᵏᵗ
Half-Life Calculations Half-life is the time it takes for half of a substance to decay. After n half-lives:
remaining amount = P₀ · (1/2)ⁿ
Example: If a radioactive isotope has a half-life of 6 hours, after 12 hours (2 half-lives) only 1/4 of the original amount remains.
Initial Value (Initial Condition) P₀ is the starting value at t = 0. It is the y-intercept of the exponential graph.
Example: In P(t) = 50 · e⁻⁰·¹ᵗ, the initial value is 50.
Interpreting Exponential Models
- The sign of the exponent determines growth (+) or decay (-)
- The rate constant k or r determines how fast the change occurs
- Doubling time (growth) and half-life (decay) summarize the speed of change
Key Points
- Exponential growth uses a base greater than 1; decay uses a base between 0 and 1
- Half-life is the time for half the substance to decay
- After n half-lives, remaining amount = P₀ · (1/2)ⁿ
- P₀ is the initial value at t = 0
- The rate constant controls how fast growth or decay occurs