Logarithmic notation is not likely something you have encountered in previous math courses. Therefore, you may be curious as to what the base e is. Sometimes called Euler's number after the 18th century Swiss mathematician, Leonhard Euler. The number e is an important mathematical constant. It is a value that arises frequently in the study of compound interest. The number e is irrational and is approximately equal to 2.71828. In your study of logarithms, remember that e is a number and not a variable.

In the following Check it Out activity, you will explore the characteristics of logarithmic functions of the form y = a logb x, where b = 10 or b = e, and a is a real number. Recall, that for b = 10, y = a log10 x is equivalent to y = a log x and for b = e, y = a loge x is equivalent to y = a ln x.

Use your graphing calculator to explore logarithmic functions.

Complete the questions that investigate characteristics of the logarithmic function.

Click here to check your answers.

      1. Use a graphing calculator to graph the logarithmic function y = log x and the exponential function y = 10x. Sketch and label both graphs on the axes shown below.

      1. How are the graphs of y = log x and y = 10x related?

    1. Each of the following functions has the form y = a log x. For each a-value, complete the chart as shown in the example.

Parameter a
Equation
Graph
8
y = 8 log x
4
0.5
-0.5
-4
-8

  1. Examine the graphs of the functions in question 3, and state the following characteristics:
  • x-intercept
  • number of y-intercepts
  • end behaviour

    \(a>0\)

    \(a<0\)

  • domain
  • range
      1. Using a graphing calculator to graph the logarithmic function y = ln x and the exponential function y = ex. Sketch and label both graphs on the axes shown below.

      1. How are the graphs of y = ln x and y = ex related?

      2. Each of the following functions has the form y = a ln x. For each a-value, complete the chart as shown in the example.
Parameter a
Equation
Graph
8
y = 8 ln x
4
0.5
-0.5
-4
-8

      1. Examine the graphs of the functions in question 7, and state the following characteristics:
    • x-intercept
    • number of y-intercepts
    • end behaviour

      \(a>0\)

      \(a<0\)

    • domain
    • range

        1. Can you predict the end behaviour of functions of the form y = a log x or y = a ln x based on the parameter a? Explain.

        2. In logarithmic functions of the form y = a logb x, did changing the base from b = 10 to b = e cause any characteristics to change? Explain.

    In Check it Out, you should have discovered that the logarithmic functions y = a log x and y = a ln x have the same characteristics.