Exponential Functions: Converting Forms and Growth Rates 5.2

Exponential and Logarithmic Functions

Converting Between Exponential Forms

Exponential functions are commonly written in two forms: $Q = ab^t$ and $Q = ae^{kt}$. Understanding how to convert between these forms is essential for analyzing growth and decay in mathematical models.

• Standard Exponential Form: $Q = ab^t$, where a is the initial value, b is the growth/decay factor, and t is time.

• Continuous Exponential Form: $Q = ae^{kt}$, where k is the continuous growth (or decay) rate.

• Conversion Formula: To convert $Q = ab^t$ to $Q = ae^{kt}$, use $k = ext{ln}(b)$, since $b^t = e^{t ext{ln}(b)}$.

Example: Convert $Q = 120(0.97)^t$ to $Q = ae^{kt}$.

• Identify $a = 120$, $b = 0.97$.

• Calculate $k = ext{ln}(0.97) \approx -0.03046$.

• So, $Q = 120e^{-0.03046t}$.

Additional info: This conversion is useful for solving problems involving continuous growth or decay, such as population models or radioactive decay.

Converting to the Form $Q = ab^t$

Sometimes, exponential functions are given in the continuous form $Q = ae^{kt}$ and need to be converted to the standard form $Q = ab^t$.

• Conversion Formula: $b = e^k$, since $e^{kt} = (e^k)^t$.

• Example: Convert $Q = 160e^{0.08t}$ to $Q = ab^t$.

• Identify $a = 160$, $k = 0.08$.

• Calculate $b = e^{0.08} \approx 1.0833$.

• So, $Q = 160(1.0833)^t$.

Additional info: This form is often used for discrete time intervals, such as annual growth rates.

Finding Initial Value, Growth Rate, and Continuous Growth Rate

Given an exponential function, you may be asked to identify the initial value, the (discrete) growth rate, and the continuous growth rate.

• Initial Value ($a$): The value of $Q$ when $t = 0$.

• Growth Rate ($r$): For $Q = ab^t$, $r = b - 1$.

• Continuous Growth Rate ($k$): For $Q = ae^{kt}$, $k = ext{ln}(b)$.

Example: For $Q = 290(1.82)^t$:

• Initial value: $a = 290$

• Growth rate: $r = 1.82 - 1 = 0.82$ (or 82%)

• Continuous growth rate: $k = ext{ln}(1.82) \approx 0.598$

Example: For $Q = 0.185(1.77)^t$:

• Initial value: $a = 0.185$

• Growth rate: $r = 1.77 - 1 = 0.77$ (or 77%)

• Continuous growth rate: $k = ext{ln}(1.77) \approx 0.570$

Summary Table: Exponential Function Forms and Conversion

Form	Parameter	Interpretation	Conversion
$Q = ab^t$	$a$	Initial value	Same in both forms
$Q = ab^t$	$b$	Growth/decay factor	$b = e^k$
$Q = ae^{kt}$	$k$	Continuous growth rate	$k = ext{ln}(b)$

Additional info: These conversions are fundamental in Precalculus for understanding exponential models in both discrete and continuous contexts.