Problem 80 - College Algebra

Question

In Exercises \(79-82,\) use a graphing utility and the change-of-base property to graph each function. $$ y=\log _{15} x $$

Step-by-Step Solution

Answer

The graph of the function \(y=\log_{15} x\) was plotted using the change-of-base formula and a graphing utility. The graph provides insight into the behavior of the function, which decreases for \(x<1\), is undefined at \(x=0\), and increases for \(x>1\).

Step 1: Understanding the Change-of-Base Formula

Recall the change-of-base formula: \(\log_b a = \frac{\log a}{\log b}\). This formula allows to convert a logarithm of any base to a fraction of two logarithms.

Step 2: Apply the Change-of-Base Formula

Next, apply the change-of-base formula to \(y=\log_{15} x\). This will get the equation \(y=\frac{\log x}{\log {15}}\). We've now transformed the original function into a format compatible with the graphing tool.

Step 3: Enter the Function into the Graphing Tool

Enter the reformulated function, \(y=\frac{\log x}{\log {15}}\), into the graphing tool. Be sure to zoom in or out and adjust the view window as necessary to best visualize the function.

Step 4: Graph the Function

Plot the graph. The function will decrease in value for \(x<1\), is undefined at \(x=0\), and will increase for \(x>1\). The graph of the function provides a visual representation of the function's behavior.

Key Concepts

Graphing Functions

When it comes to graphing functions, the primary goal is to get a visual understanding of how a function behaves. Graphs allow us to see the general form and nature of the relationships between variables, which can often be less apparent through equations alone. For instance:

• The slope of a graph can indicate whether a function is increasing or decreasing.
• Intercepts show where the function crosses the axes.
• Asymptotes can reveal horizontal or vertical lines that the graph approaches but never quite touches.

With graphing utilities, you can input complex functions and see their graphs instantly. This makes it easier to comprehend the effects of parameters within the function. Adjusting the viewing window or using specific tools within the graphing software can often help in providing a more detailed picture.
Graphing functions involves understanding their basic characteristics and behavior over different intervals. Be sure to recognize where a function is defined and where it isn't, as this will often affect how you plot the graph.

Logarithm Base Conversion

Logarithm base conversion is a valuable tool when graphing or simplifying logarithmic expressions. The change-of-base formula is the key tool for converting logarithms from one base to another. Understanding this formula is crucial because calculators typically only compute logarithms in base 10 or base e (natural logarithms).The formula looks like this:\[\log_b a = \frac{\log_k a}{\log_k b}\]where:

• \(b\): the original base of the logarithm you want to change
• \(k\): a new base like 10 or \(e\)
• \(a\): the value you’re taking the logarithm of

In practical terms, when turning the equation \(y=\log_{15} x\) into a graphable format, you apply this formula to get \(y=\frac{\log x}{\log 15}\).
This way, you can input it into graphing instruments without encountering issues concerning unsupported log bases. This change-of-base formula makes it possible to work with logarithmic functions efficiently using standard scientific calculators as well.

Logarithmic Functions

Logarithmic functions have unique attributes that differ from polynomial, exponential, and other functions. They involve inverses of exponential functions, turning multiplication into addition. This means logarithmic functions grow at a much slower rate than exponentials.Some key features of logarithmic functions include:

• They are undefined for non-positive values of \(x\) because the logarithm of zero or a negative number is not a real number.
• The function \(y=\log_b x\) will pass through the point \( (1, 0) \) because the log of 1 is always 0, irrespective of the base.
• As \(x\) increases, the function grows logarithmically, meaning slower as \(x\) gets larger.

When you plot a logarithmic function after using the change-of-base formula, you'll notice the curve’s characteristic, moving gradually upwards through one quadrant. The visual portrayal can help in understanding limits, asymptotic behavior, and the general trend of change.
Working with logarithmic functions involves adjusting to their particular rules and behavior, but their real-world applications are vast, from measuring earthquakes to calculating compound interest!