Problem 79 - College Algebra

Question

In Exercises \(79-82,\) use a graphing utility and the change-of- base property to graph each function. $$ y=\log _{3} x $$

Step-by-Step Solution

Answer

The graph of \(y=\log_{3}x\) is the graph of \(\frac{\log_{10}x}{\log_{10}3}\).

Step 1: Understand the Change of Base Formula

The base change formula is \(\log_b a = \frac{\log_d a}{\log_d b}\), where \(b\) is the base of the original logarithm, \(a\) is the argument of the logarithm, and \(d\) is the new base. This formula allows us to change the base of a logarithm.

Step 2: Apply the Change of Base Formula

Apply the change of base formula to the given function \(y=\log_{3} x\). Let us change the base from 3 to 10. This can be done by applying the formula: \[y = \frac{\log_{10}x}{\log_{10}3}\]

Step 3: Graph the function

Plot the function using a graphing utility by entering \(\frac{\log_{10}x}{\log_{10}3}\) into the utility. Usually, the x-axis represents the values of x and the y-axis represents the values of the function.

Key Concepts

Change of Base Formula

The change of base formula is a fundamental tool when working with logarithms. It is frequently used to simplify calculations or when graphing logarithmic functions. The essence of this formula is that it allows you to convert logarithms from one base to another, making them easier to manipulate with common bases such as 10 or e.
To use the change of base formula, you apply: \( \log_b a = \frac{\log_d a}{\log_d b} \). Here, \( b \) is the original base, \( a \) is the argument of the logarithm, and \( d \) is the base to which you wish to convert.
For example, if you have \( \log_3 x \), you can convert it to base 10 by using the formula: \[ \log_3 x = \frac{\log_{10} x}{\log_{10} 3} \] This form makes it possible to evaluate logarithms using calculators or graphing utilities, which typically do not support arbitrary bases without conversion.

Graphing Utility

Graphing utilities, such as handheld calculators or software like Desmos, are invaluable tools for visualizing mathematical functions. They allow you to plot functions over specific ranges, giving insight into the behavior of the function.
When dealing with functions like \( y = \frac{\log_{10} x}{\log_{10} 3} \), graphing utilities simplify the process by bypassing the manual calculation of each log value. Instead, you just input the converted formula, and the tool displays the graph for you.
This is particularly useful:

• For observing how logarithmic functions behave as \( x \) increases or decreases.
• For identifying key characteristics like intercepts, asymptotes, and growth tendencies.
• For comparing against other transformations like shifts and scalings.

By visualizing \( y = \log_3 x \) in a graphing utility using its base 10 representation, you gain intuitive understanding quickly and accurately.

Base 10 Logarithms

Logarithms are often expressed in base 10, known as "common logarithms," denoted as \( \log_{10} \). Base 10 logarithms are prevalent due to their integration with the decimal number system.
This format greatly simplifies complex calculations because most scientific calculators and graphing utilities support only base 10 and base e (natural) logarithms natively.
Using base 10 is:

• Efficient: Calculations involving exponents or powers that are multiples of ten become straightforward.
• Accessible: Almost all calculators provide a direct function for \( \log_{10} \) without needing to change settings.
• Suitable for science and engineering: Since many real-world applications involve scaling factors of 10.

When graphing a function like \( y = \log_3 x \), converting it to a base 10 format with the change of base formula not only makes it practical for plotting but also leverages computational efficiencies, hence being more aligned with typical technological capabilities students use.