Problem 8 - Thinking Mathematically

Question

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 \(y\). \(y=\log _{5} x\)

Step-by-Step Solution

Answer

The exponential form of \(y = \log_5 x\) is \(5^y = x\). After substituting selected y-values (-2, -1, 0, 1, 2) into the exponential equation, we obtain respective x-values of (0.04, 0.2, 1, 5, 25), which are used to graph the function.

Step 1: Convert to Exponential Form

The general formula for conversion from logarithmic form to exponential form is \(\log_b a = n \) equivalent to \(b^n = a\). Using this formula, the exponential form of the equation \(y = \log_5 x\) will be \(5^y = x\).

Step 2: Create a Table of Coordinates

Select the values of \(y = -2, -1, 0, 1, 2\) and substitute them into the equation \(5^y = x\) to get the respective 'x' values. This will give a table like this: \[ \begin{tabular}{|c|c|} \n\hline \n y & x \\ \n\hline \n-2 & 0.04 \\ \n-1 & 0.2 \\ \n 0 & 1 \\ \n 1 & 5 \\ \n 2 & 25 \\ \n\hline \n\end{tabular} \]

Step 3: Graph the Function

Plot the points from the table of coordinates on the graph. Each pair (x, y) represents a point on the graph. Connect the points to get the graph of the function \(y = \log_5 x\) which shows a typical shape of a logarithmic function. Remember that the graph will not extend to negative 'x' values as logarithms are undefined for negative numbers.

Key Concepts

Exponential Form

Understanding the conversion from logarithmic to exponential form is crucial for solving logarithmic equations. The traditional logarithmic equation is written as

• \(\log_b a = n\),

where \(b\) is the base of the logarithm, \(a\) is the given value, and \(n\) is the exponent. This can be converted into its equivalent exponential form:

• \(b^n = a\).

In simpler terms, what this means is that the base raised to the power of the exponent equals the given value.

For the specific equation given in the exercise, \(y = \log_5 x\):

• \(5^y = x\).

This shows that the base \(5\) raised to the power \(y\) will give us \(x\). By rewriting it in exponential form, it becomes easier to solve for \(x\) when different values of \(y\) are provided. This is a vital step to create a useful coordinate table and, eventually, to plot the graph.

Coordinate Table

A coordinate table is an organized way to collect pairs of input and output values, which makes it easier to plot a graph. In this exercise, the table is formed from the exponential form of the logarithmic function.

• Start by selecting specific values for the variable \(y\).
• Here, these values were chosen as \(-2, -1, 0, 1, 2\).
• They represent various points on the y-axis.

By substituting each value of \(y\) into the formula \(5^y = x\), we derive corresponding values for \(x\):

• For \(y = -2\), \(5^{-2} = 0.04\) which is \(x\).
• For \(y = -1\), \(5^{-1} = 0.2\).
• For \(y = 0\), \(5^0 = 1\).
• For \(y = 1\), \(5^1 = 5\).
• For \(y = 2\), \(5^2 = 25\).

This process results in coordinate pairs of \((x, y)\) which can then be plotted on a graph to visualize the function.

Remember, creating a table helps in visualizing how the function behaves and changes with different inputs.

Graphing Functions

Graphing a function involves plotting coordinate pairs on a graph to visualize the relationship between variables. Here, the focus is on the logarithmic function \(y = \log_5 x\).

• Begin by plotting the points from the table of coordinates, calculated in the previous step.
• Examples of these pairs include \((0.04, -2), (0.2, -1), (1, 0), (5, 1), (25, 2)\).

Each point represents a location on the graph where the function takes a specific value.

Once the points are plotted:

• Connect them in a smooth curve.
• The graph reveals the typical shape of a logarithmic curve: it passes through the point (1, 0) and approaches the x-axis asymptotically as x becomes smaller, indicating it never crosses the x-axis.
• It also rises sharply as \(x\) increases.

It is important to note that the graph only exists for positive \(x\) values since logarithms are undefined for negative or zero \(x\).

By visualizing the graph, you can appreciate the nature of logarithmic functions and their exponential relationships with their inverse functions.