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Study Guides > College Algebra

Key Concepts & Glossary

Key Equations

Half-life formula If [latex]\text{ }A={A}_{0}{e}^{kt}[/latex], < 0, the half-life is [latex]t=-\frac{\mathrm{ln}\left(2\right)}{k}[/latex].
Carbon-14 dating [latex]t=\frac{\mathrm{ln}\left(\frac{A}{{A}_{0}}\right)}{-0.000121}[/latex].[latex]{A}_{0}[/latex] A is the amount of carbon-14 when the plant or animal died t is the amount of carbon-14 remaining today is the age of the fossil in years
Doubling time formula If [latex]A={A}_{0}{e}^{kt}[/latex], > 0, the doubling time is [latex]t=\frac{\mathrm{ln}2}{k}[/latex]
Newton’s Law of Cooling [latex]T\left(t\right)=A{e}^{kt}+{T}_{s}[/latex], where [latex]{T}_{s}[/latex] is the ambient temperature, [latex]A=T\left(0\right)-{T}_{s}[/latex], and k is the continuous rate of cooling.

Key Concepts

  • The basic exponential function is [latex]f\left(x\right)=a{b}^{x}[/latex]. If > 1, we have exponential growth; if 0 < < 1, we have exponential decay.
  • We can also write this formula in terms of continuous growth as [latex]A={A}_{0}{e}^{kx}[/latex], where [latex]{A}_{0}[/latex] is the starting value. If [latex]{A}_{0}[/latex] is positive, then we have exponential growth when > 0 and exponential decay when < 0.
  • In general, we solve problems involving exponential growth or decay in two steps. First, we set up a model and use the model to find the parameters. Then we use the formula with these parameters to predict growth and decay.
  • We can find the age, t, of an organic artifact by measuring the amount, k, of carbon-14 remaining in the artifact and using the formula [latex]t=\frac{\mathrm{ln}\left(k\right)}{-0.000121}[/latex] to solve for t.
  • Given a substance’s doubling time or half-time, we can find a function that represents its exponential growth or decay.
  • We can use Newton’s Law of Cooling to find how long it will take for a cooling object to reach a desired temperature, or to find what temperature an object will be after a given time.
  • We can use logistic growth functions to model real-world situations where the rate of growth changes over time, such as population growth, spread of disease, and spread of rumors.
  • We can use real-world data gathered over time to observe trends. Knowledge of linear, exponential, logarithmic, and logistic graphs help us to develop models that best fit our data.
  • Any exponential function with the form [latex]y=a{b}^{x}[/latex] can be rewritten as an equivalent exponential function with the form [latex]y={A}_{0}{e}^{kx}[/latex] where [latex]k=\mathrm{ln}b[/latex].


carrying capacity
in a logistic model, the limiting value of the output
doubling time
the time it takes for a quantity to double
the length of time it takes for a substance to exponentially decay to half of its original quantity
logistic growth model
a function of the form [latex]f\left(x\right)=\frac{c}{1+a{e}^{-bx}}[/latex] where [latex]\frac{c}{1+a}[/latex] is the initial value, c is the carrying capacity, or limiting value, and b is a constant determined by the rate of growth
Newton’s Law of Cooling
the scientific formula for temperature as a function of time as an object’s temperature is equalized with the ambient temperature
order of magnitude
the power of ten, when a number is expressed in scientific notation, with one non-zero digit to the left of the decimal

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  • Precalculus. Provided by: OpenStax Authored by: Jay Abramson, et al.. Located at: https://openstax.org/books/precalculus/pages/1-introduction-to-functions. License: CC BY: Attribution. License terms: Download For Free at : http://cnx.org/contents/[email protected]..