Exponents and exponential growth/decay Knowledge-Based Study

Study notes

1. Laws and Properties of Exponents

   An exponent represents repeated multiplication of a base number by itself. For any real number base a and positive integer exponent n:

   $$a^n = a \cdot a \cdot a \dots a \quad (n\text{ factors of } a)$$

   Working with exponential functions requires mastery of key exponent rules:

   • Product Rule: $a^m \cdot a^n = a^{m+n}$
   • Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$, provided $a \neq 0$
   • Power of a Power Rule: $(a^m)^n = a^{m \cdot n}$
   • Zero Exponent Rule: $a^0 = 1$, for all $a \neq 0$
   • Negative Exponent Rule: $a^{-n} = \frac{1}{a^n}$, for all $a \neq 0$
   • Rational Exponent Rule: $a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$, where $n > 0$

   Defining the Exponential Function

   An exponential function is a mathematical relationship of the form:

   $$f(x) = a \cdot b^x$$

   where:

   • a is a non-zero real constant representing the initial value or vertical intercept $(f(0) = a)$.
   • b is the base, a positive real constant such that $b > 0$ and $b \neq 1$.
   • x is the independent variable, which can take any real value.

   The behavior of $f(x)$ depends fundamentally on the value of the base b:

   • If $b > 1$, the outputs increase as $x$ increases, demonstrating exponential growth.
   • If $0 < b < 1$, the outputs decrease as $x$ increases, demonstrating exponential decay.

   All basic exponential functions of the form $f(x) = b^x$ pass through the point $(0, 1)$ and have a horizontal asymptote along the line $y = 0$.

   Modeling Growth Over Time

   Exponential growth occurs when a quantity increases by a constant percentage rate across equal increments of time.

   The standard model for exponential growth is given by:

   $$A(t) = A_0(1 + r)^t$$

   In this equation:

   • $A(t)$ is the amount at time $t$.
   • $A_0$ is the initial amount at $t = 0$.
   • $r$ is the fractional growth rate per time period, with $r > 0$.
   • $1 + r$ is the growth factor, equivalent to base $b$ where $b > 1$.
   • $t$ represents the number of elapsed time periods.

   Example Application

   A bacterial culture initially contains 500 cells and increases by 12% each hour. To determine the population after 4 hours:

   $$A(4) = 500(1 + 0.12)^4 = 500(1.12)^4 \approx 500(1.5735) \approx 787\text{ cells}$$

   Modeling Decay Over Time

   Exponential decay occurs when a quantity decreases by a constant percentage rate across equal increments of time.

   The standard model for exponential decay is given by:

   $$A(t) = A_0(1 - r)^t$$

   In this equation:

   • $A(t)$ is the remaining quantity at time $t$.
   • $A_0$ is the initial quantity at $t = 0$.
   • $r$ is the fractional decay rate per time period, with $0 < r < 1$.
   • $1 - r$ is the decay factor, equivalent to base $b$ where $0 < b < 1$.
   • $t$ represents elapsed time periods.

   Half-Life Formulation

   A specialized form of exponential decay is radioactive decay based on half-life ($h$), the duration required for half of the substance to disintegrate:

   $$A(t) = A_0 \left(\frac{1}{2}\right)^{t/h}$$

   Here, the base is $b = 0.5$, reflecting a halving of the quantity every $h$ time units.

   Distinguishing Growth and Decay

   A summary of the differences between exponential growth and exponential decay models is given below:

   • Growth Factor vs. Decay Factor: In $f(t) = a \cdot b^t$, $b = 1 + r > 1$ indicates growth, whereas $b = 1 - r < 1$ indicates decay.
   • Rate of Change: For both models, the absolute rate of change $\frac{dA}{dt}$ is directly proportional to the current quantity $A(t)$, leading to rapid acceleration in growth and asymptotic leveling in decay.
   • Long-Term Behavior (End Behavior): As $t \to \infty$, exponential growth functions tend toward $+\infty$, while exponential decay functions approach the horizontal asymptote $A = 0$.

   Recognizing whether the base is greater than 1 or between 0 and 1 allows immediate identification of the model type.