Problem 7 - Thinking Mathematically

Question

In Exercises 7-8, a. Rewrite each equation in exponential form. b. Use a table of coordinates and the exponential form from part (a) to graph each logarithmic function. Begin by selecting \(-2,-1,0,1\), and 2 for \(y\). \(y=\log _{4} x\)

Step-by-Step Solution

Answer

The exponential form of \(y=\log _{4} x\) is \(4^y = x\). The table of coordinates contains: (\(-2, 0.0625\); \(-1, 0.25\); \(0, 1\); \(1, 4\); \(2, 16\)). The graph of this function would show these points and a curve increasing as x increases, passing through the point (1, 0) and approaching but never crossing the y-axis.

Step 1: Convert logarithmic equation to exponential form

The general formula of a logarithm is \(\log_{b} a = n\), which is equivalent to \(b^n = a\). So, applying this to \(y=\log _{4} x\) we get \(4^y = x\).

Step 2: Create a coordinates table

To create the coordinates table, the given y-values (-2, -1, 0, 1, 2) will be plugged into the exponential equation \(4^y = x\). Calculate the corresponding x-value for each y-value to obtain the coordinates: \nFor \(y = -2, x = 4^{-2} = 0.0625\), \nFor \(y = -1, x = 4^{-1} = 0.25\), \nFor \(y = 0, x = 4^{0} = 1\), \nFor \(y = 1, x = 4^{1} = 4\), \nFor \(y = 2, x = 4^{2} = 16\).

Step 3: Graph the logarithmic function

With the table of coordinates (\(-2, 0.0625\); \(-1, 0.25\); \(0, 1\); \(1, 4\); \(2, 16\)), plot these points on a graph. The y-axis represents the y-values and the x-axis represents the x-values. Remember that the graph of the logarithmic function \(\log _{b} x\) is a smooth curve that increases as x increases. It passes through the point (1, 0) and approaches the y-axis but never crosses it (that is, it has a vertical asymptote at \(x=0\)).

Key Concepts

Converting Logarithms to Exponentials

Transforming a logarithmic equation into its exponential form is a foundational skill in understanding how these functions relate to each other. Take the general logarithmic equation: \(\log_{b} a = n\). This equation means that the base \(b\), raised to the power of \(n\), equals \(a\). In practical terms, if we have \(y=\log _{4} x\), we can rewrite this in exponential form as \(4^y = x\). This step is critical because it allows us to work with the equation algebraically and understand the relationship between the variables in a more intuitive way.

Creating a Table of Coordinates

To visualize a logarithmic function, it's helpful to create a table of coordinates by selecting values for \(y\) and calculating the corresponding \(x\) values using the exponential form obtained from converting the logarithmic equation. For example, if we have our exponential equation \(4^y = x\), we can plug in selected \(y\) values like -2, -1, 0, 1, and 2 to find the corresponding \(x\) values:

• For \(y = -2\), \(x = 4^{-2} = 0.0625\)
• For \(y = -1\), \(x = 4^{-1} = 0.25\)
• For \(y = 0\), \(x = 4^{0} = 1\)
• For \(y = 1\), \(x = 4^{1} = 4\)
• For \(y = 2\), \(x = 4^{2} = 16\)

This table serves as the basis for plotting the function on a graph.

Understanding Exponential and Logarithmic Relationships

Grasping the relationship between exponential and logarithmic functions is crucial for decoding how they influence each other. They are essentially inverses. An exponential function like \(b^y = x\) involves a constant base \(b\) raised to a variable exponent \(y\) to produce the output \(x\), whereas a logarithmic function like \(\log_{b} x = y\) essentially asks the question: 'To what power must we raise \(b\) to obtain \(x\)?' The answer to that question is the value of \(y\). The exponential form is useful for finding coordinates to plot, while the logarithmic form gives us insights into the growth patterns of various phenomena, particularly where growth occurs in multiplicative steps.

Plotting Points of Logarithmic Functions

Once we have our table of coordinates, plotting points of a logarithmic function becomes an exercise in connecting the dots methodically to reveal the characteristic curve of the graph. Each pair of \(x\) and \(y\) corresponds to a point on the Cartesian plane. For \(y=\log _{4} x\), the points derived from our table would be (\(0.0625 -2\)), (\(0.25 -1\)), (\(1 0\)), (\(4 1\)), (\(16 2\)). It is important to remember that logarithmic functions have distinctive features; the curve is always increasing and passes through the point \(1, 0\), and they have a vertical asymptote at \(x=0\) which the curve approaches but never crosses. These properties help us to draw the graph accurately.