Why It Matters: Exponential and Logarithmic Functions
Why It Matters: Exponential and Logarithmic Functions
Many sources credit Albert Einstein as saying, “Compound interest is the eighth wonder of the world. He who understands it, earns it ... he who doesn't ... pays it.” You probably already know this if you have ever invested in an account or taken out a loan. Interest is the amount added to the balance. The beauty, in the case of investing, is that once interest is earned, it also earns interest. This idea of interest earning interest is known as compound interest. (It isn’t quite as beautiful on money you owe.) Interest can be compounded over different time intervals. It might be compounded annually (once per year), or more often, such as semi-annually (twice per year), quarterly (four times per year), monthly (12 times per year), weekly (52 times per year), or daily (365 times per year). There is also one more option—compounded continuously—which is the theoretical concept of adding interest in infinitesimally small increments. Although not actually possible, it provides the limit of compounding and is therefore a useful quantity in economics.
Suppose you inherit $10,000. You decide to invest in in an account paying 3% interest compounded continuously. What will the balance be in 5 years, 10 years, or even 50 years? You’ll want to know, especially for retirement planning.
In this module, you will learn about the function you can evaluate to answer these questions. You will also discover how to make changes to the variables involved, such as time or initial investment, to alter your results.
Learning Outcomes
Review for Success- Use the properties of exponents to rewrite expressions containing exponents.
- Evaluate and solve functions.
- Identify the base of an exponential function and restrictions for its value.
- Find the equation of an exponential function.
- Use the compound interest formula.
- Evaluate exponential functions with base e.
- Given two data points, write an exponential function.
- Identify initial conditions for an exponential function.
- Find an exponential function given a graph.
- Use a graphing calculator to find an exponential function.
- Find an exponential function that models continuous growth or decay.
- Determine whether an exponential function and its associated graph represents growth or decay.
- Sketch a graph of an exponential function.
- Graph exponential functions shifted horizontally or vertically and write the associated equation.
- Graph a stretched or compressed exponential function.
- Graph a reflected exponential function.
- Write the equation of an exponential function that has been transformed.
- Convert from logarithmic to exponential form.
- Convert from exponential to logarithmic form.
- Evaluate logarithms with and without a calculator.
- Evaluate logarithms with base 10 and base e.
- 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.
Licenses & Attributions
CC licensed content, Original
- Why It Matters: Exponential and Logarithmic Functions. Authored by: Lumen Learning. License: CC BY: Attribution.
CC licensed content, Shared previously
- Piggy bank on dollar bills. Authored by: Pictures of Money. Located at: https://www.flickr.com/photos/pictures-of-money/17299241862/. License: CC BY: Attribution.
