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Study Guides > College Algebra CoRequisite Course

Why It Matters: Exponential and Logarithmic Equations and Models

Why It Matters: Exponential and Logarithmic Equations

Image shows a man’s hand holding a fossil outside near water. You have a job assisting an archeologist who has just discovered a fossil that appears to be an animal bone. She assigns you the task of determining how old the bone is. Where do you start? Fortunately, you know that living things contain a radioactive form of carbon called carbon-14. Like all radioactive elements, carbon-14 decays at a predictable rate known as its half-life. The half-life of a radioactive element is the amount of time required for half of a sample to decay. The half-life of carbon-14 is 5,730 years. Given an original sample of carbon-14 of 100g, the table shows the mass remaining after each half-life.
Amount of sample (g) 100 50 25 12.5 6.25
Time (years) 0 5730 11,460 17,190 22,920
But, what if the bone started with a different mass of carbon-14 or a different number of years has passed? To better study the bone, you need to know that the rate of decay of a radioactive element can be modeled with an exponential function. Given a couple of data points, you can build a model that represents the decay of carbon over time for your specimen. As you complete this module, keep the following questions in mind. Then at the end of the module, we will return to develop a model for the decay of carbon-14.
  • How do you develop a model for the decay of carbon-14?
  • How can you use the model to determine the amount of carbon-14 that remains after any number of years?
  • What would a graph of the decay of carbon-14 look like?

Learning Outcomes

Review for Success
  • Define the identity and zero properties of exponents and logarithms.
  • Define the inverse property of exponents and logarithms.
  • Define the one-to-one properties of exponents and logarithms.
  • Define the product and quotient properties of exponents and logarithms.

Properties of Logarithms

  • Rewrite a logarithmic expression using the power rule, product rule, or quotient rule.
  • Expand logarithmic expressions using a combination of logarithm rules.
  • Condense logarithmic expressions using logarithm rules.
  • Expand a logarithm using a combination of logarithm rules.
  • Condense a logarithmic expression into one logarithm.
  • Rewrite logarithms with a different base using the change of base formula.

Exponential and Logarithmic Equations

  • Solve an exponential equation with a common base.
  • Rewrite an exponential equation so all terms have a common base then solve.
  • Recognize when an exponential equation does not have a solution.
  • Use logarithms to solve exponential equations.
  • Solve a logarithmic equation algebraically.
  • Solve a logarithmic equation graphically.
  • Use the one-to-one property of logarithms to solve a logarithmic equation.
  • Solve a radioactive decay problem.

Exponential and Logarithmic Models

  • Graph exponential growth and decay functions.
  • Solve problems involving radioactive decay, carbon dating, and half life.
  • Use Newton's Law of Cooling.
  • Use a logistic growth model.
  • Choose an appropriate model for data.
  • Use a graphing utility to create an exponential regression from a set of data.
 

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  • Why It Matters: Exponential and Logarithmic Equations and Models. Authored by: Lumen Learning. License: CC BY: Attribution.

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