Uninhibited growth can be modeled by exponential functions...

Uninhibited growth can be modeled by exponential functions other than $A(t)=A_{0} e^{k t} .$ For example, if an initial population $P_{0}$ requires $n$ units of time to double, then the function $P(t)=P_{0} \cdot 2^{t / n}$ models the size of the population at time t. Likewise, a population requiring $n$ units of time to triple can be modeled by $P(t)=P_{0} \cdot 3^{t / n}$. An insect population grows exponentially. (a) If the population triples in 20 days, and 50 insects are present initially, write an exponential function of the form $P(t)=P_{0} \cdot 3^{t / n}$ that models the population. (b) What will the population be in 47 days? (c) When will the population reach $700 ?$ (d) Express the model from part (a) in the form $A(t)=A_{0} e^{k t}$

Uninhibited growth can be modeled by exponential functions other than $A(t)=A_{0} e^{k t} .$ For example, if an initial population $P_{0}$ requires $n$ units of time to double, then the function $P(t)=P_{0} \cdot 2^{t / n}$ models the size of the population at time t. Likewise, a population requiring $n$ units of time to triple can be modeled by $P(t)=P_{0} \cdot 3^{t / n}$.
An insect population grows exponentially.
(a) If the population triples in 20 days, and 50 insects are present initially, write an exponential function of the form $P(t)=P_{0} \cdot 3^{t / n}$ that models the population.
(b) What will the population be in 47 days?
(c) When will the population reach $700 ?$
(d) Express the model from part (a) in the form $A(t)=A_{0} e^{k t}$

Key Concepts

Exponential Growth

Exponential growth occurs when the rate of increase of a quantity is proportional to the current amount. In modeling populations, this means that the population grows faster as it becomes larger, and the growth can be represented as an exponential function where time is the independent variable.

Doubling/Tripling Time in Exponential Models

This concept involves using a specific time period in which a quantity multiplies by a fixed factor, such as doubling or tripling. It allows one to calibrate the exponential function by relating real-world time intervals to the base of the exponential, simplifying the understanding and calculation of growth over time.

Conversion Between Exponential Forms

Sometimes it is useful to express exponential models in different but equivalent forms. Converting a model from a base like 3 to the natural exponential function involves using logarithmic identities. This conversion, where k = ln(3)/n, provides flexibility in analyzing and solving problems with exponential growth.

Application of Logarithms in Solving Exponential Equations

Logarithms are used to solve for variables when they are located in the exponent of an exponential equation. They allow us to 'bring down' the exponent by converting multiplicative relationships into additive ones, thereby facilitating the algebraic manipulation and solution of the equation for time or growth rate.

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