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Study Guides > College Algebra CoRequisite Course

Introduction to Graphs of Logarithmic Functions

Learning Outcomes

By the end of this lesson, you will be able to:
  • Determine the domain and range of a logarithmic function.
  • Determine the x-intercept and vertical asymptote of a logarithmic function.
  • Identify whether a logarithmic function is increasing or decreasing and give the interval.
  • Identify the features of a logarithmic function that make it an inverse of an exponential function.
  • Graph horizontal and vertical shifts of logarithmic functions.
  • Graph stretches and compressions of logarithmic functions.
  • Graph reflections of logarithmic functions.
Previously, we saw how creating a graphical representation of an exponential model gives us another layer of insight for predicting future events. How do logarithmic graphs give us insight into situations? Because every logarithmic function is the inverse function of an exponential function, we can think of every output on a logarithmic graph as the input for the corresponding inverse exponential equation. In other words, logarithms give the cause for an effect. To illustrate, suppose we invest $2500 in an account that offers an annual interest rate of 5% compounded continuously. We already know that the balance in our account for any year t can be found with the equation [latex]A=2500{e}^{0.05t}[/latex]. What if we wanted to know the year for any balance? We would need to create a corresponding new function by interchanging the input and the output; thus we would need to create a logarithmic model for this situation. By graphing the model, we can see the output (year) for any input (account balance). For instance, what if we wanted to know how many years it would take for our initial investment to double? The graph below shows this point. A graph titled, In this section we will discuss the values for which a logarithmic function is defined and then turn our attention to graphing the family of logarithmic functions.

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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]..
  • College Algebra. Provided by: OpenStax Authored by: Abramson, Jay et al.. License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].