Essential Concepts

• Logarithms are mathematical functions that reverse the effect of exponentials, with the common logarithm ([latex]log(x)[/latex]) specifically undoing the exponential [latex]10^x[/latex].
• The properties of logarithms, such as the product, quotient, and power rules, are essential for rewriting and solving equations involving exponents.
• Solving exponential equations with logarithms involves isolating the exponential term, taking the logarithm of both sides, and using the exponent property of logs to solve for the variable.
• The carrying capacity, represents the maximum population that an environment can sustain. It’s a crucial parameter in modeling logistic growth, affecting how quickly a population approaches its limit.
• Logistic growth describes a situation where growth is initially exponential but slows as it approaches a maximum limit, known as the carrying capacity. This model is more realistic for populations in constrained environments, like fish in a lake, where factors like resources and space limit growth.
• The logistic growth model can be applied to real-world situations, such as predicting future populations of animals in a forest or plants in a field, considering both the growth rate and the environment’s carrying capacity.
• The logistic growth model can be expressed in both recursive and explicit forms. The recursive form is useful for discrete growth with specific breeding times, while the explicit form is better for continuous growth scenarios.
• The growth rate, [latex]r[/latex], in logistic models can be determined using real-world data. For example, by observing the growth of a population over a set period, the rate can be calculated and used to predict future growth.
• Evaluating a logistic model involves substituting a time value to find the population at that time. Solving a logistic model means finding the time it takes for the population to reach a certain level. Both processes require an understanding of the logistic equation and its parameters.
• Exponential growth is a type of non-linear growth where a quantity increases in proportion to itself. This is common in scenarios like population growth, interest in banking, and radioactive decay.
• Exponential regression is used to fit an exponential function to a set of data, often using software tools.
• The exponential growth formula [latex]P_n =P_0(1+r)^n[/latex] can be compared and converted to the continuous growth formula [latex]y=ae^{rx}[/latex].  
• Using tools like spreadsheets, one can perform exponential regression on real-world data, such as population growth. This involves plotting data, fitting an exponential trendline, and interpreting the results, including the coefficient of determination [latex]r^2[/latex].
•  When modeling data, it’s important to compare exponential and linear models to determine which provides a better fit. This involves analyzing the [latex]r^2[/latex] values and understanding the implications of using one model over the other.
• Exponential models can be used to make predictions about future values, but it’s important to remember that these are approximations and should be used with an understanding of their limitations, especially when extrapolating beyond the available data.
• Logistic growth models describe real-world situations where growth is initially exponential but slows as it approaches a maximum limit, known as the carrying capacity.
• The logistic growth model with carrying capacity [latex]K[/latex] and growth rate [latex]r[/latex] is represented by the formula [latex]{{P}_{n}}={{P}_{n-1}}+r\left(1-\frac{{{P}_{n-1}}}{K}\right){{P}_{n-1}}[/latex] or discrete growth, and [latex]P_{t}=\dfrac{c}{1+\left(\dfrac{c}{P_{0}}-1\right)e^{-rt}}[/latex] for continuous growth.
• These models are used to represent populations that grow discretely with one breeding time per year, as well as those that breed year-round.
• Logistic growth models can be evaluated for a given input [latex]t[/latex] to make predictions about population growth over time.
• The growth rate [latex]r[/latex] can be determined using information about initial growth, and the model can then be used to evaluate the population at a specific time or solve for the time it takes to reach a certain population size.

Glossary

carrying capacity

the largest population that an environment can support

exponential regression

the process of fitting an exponential function to a set of data

logistic growth

If a population is growing in a constrained environment with carrying capacity [latex]K[/latex], and absent constraint would grow exponentially with growth rate [latex]r[/latex], then the population behavior can be described by the logistic growth model

Key Equations

continuous growth

[latex]y=ae^{x}[/latex]

continuous logistic model

[latex]P_{t}=\dfrac{c}{1+\left(\dfrac{c}{P_{0}}-1\right)e^{-rt}}[/latex]

exponent property of logarithms

[latex]\log\left({{A}^{r}}\right)=r\log\left(A\right)[/latex]

exponential growth explicit form

[latex]P_{n}=P_{0}(1+r)^{n}[/latex]

exponential growth recursive form

[latex]P­_{n}= P_{­n-1} + r P_{­n-1}[/latex]

logistic growth model

[latex]{{P}_{n}}={{P}_{n-1}}+r\left(1-\frac{{{P}_{n-1}}}{K}\right){{P}_{n-1}}[/latex]