Problem 79 - Algebra and Trigonometry

Question

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 a curve that cuts the x-axis at x = 1 and increases towards positive infinity. This is the common behavior of logarithmic functions. Their domain is (0, +\infty), range is (-\infty, +\infty), and has an x-intercept at x=1.

Step 1: Change the base of logarithm

Convert the given logarithm from base 3 to base 10 using the change of base formula. Thus, \(y=\log_3 x\) becomes \( y= \frac{ \log_{10}x }{ \log_{10}3 }\).

Step 2: Graph the function

Plot the function \( y= \frac{\log_{10}x}{\log_{10}3} \) using graphing utility software. The x-axis will represent x-values and the y-axis will represent y-values calculated from the equation.

Step 3: Understand the plot

The graph should cut the x-axis at x = 1. The graph increases as x approaches from 0 to positive infinity along the x-axis.

Key Concepts

Change of Base Formula

The change of base formula is a valuable tool when working with logarithms, especially with bases that are not commonly found on most calculators or graphing utilities. This method simplifies complex logarithmic functions, making it easier to calculate and graph them by converting to a more familiar base, usually base 10 or base e.
This conversion process involves the formula:

• If you have a logarithm in the form of \(\log_b a\), you can convert it to base 10 with the formula \(\frac{\log_{10} a}{\log_{10} b}\).

In practice, this means you take the logarithm of your number (a) and your base (b) individually using base 10 and then divide the two results. This process allows for easy computation using standard scientific calculators, as most work by default in base 10. Remember, this is a universal conversion technique that can be used for any logarithmic function, whether you're graphing or evaluating specific values.

Graphing Utility

A graphing utility is a powerful tool that helps visualize mathematical functions, including complex logarithmic ones like those involving base changes. These utilities come in different forms, such as apps, online platforms, or standalone calculators. They assist students and professionals by offering a visual representation of equations and data.
Using a graphing utility, you can do more than just plot the general shape of a function. It provides interactive features:

• Zoom and pan across different sections of the graph
• Add multiple functions at once to compare shapes and values
• Trace points along the curve to read coordinates easily

For logarithmic functions like \(y = \frac{\log_{10}x}{\log_{10}3}\), a graphing utility will show the curve as it rises from left to right, starting from 1 along the x-axis. This visual approach can aid in understanding the growth and behavior of the function in real contexts.

Base 10 Logarithm

Base 10 logarithms, often referred to as common logarithms, are fundamental in various scientific and engineering applications. Represented as \(\log_{10} x\) or simply \(\log x\), these logarithms simplify the expression and computation of logarithmic functions, especially in the context of exponential growth or decay.
Why base 10? It aligns with the decimal system, making it intuitive and straightforward for calculations. Common calculators and graphing tools utilize base 10 for quick computation:

• Facilitates easier estimation of orders of magnitude
• Simplifies the scaling of data involving different magnitudes

When transforming a function such as \(\log_3 x\) into a base 10 logarithm using the change of base formula, you harness these advantages of simplicity and commonly available tools, particularly when plotting and analyzing data graphically.

Plotting Functions

Plotting functions involves the graphical representation of mathematical equations on an x and y-axis to visualize their behavior. This process is especially useful for functions like logarithms where changes in the graph can convey significant insights.
Steps for plotting a function such as \( y = \frac{\log_{10}x}{\log_{10}3} \) include:

• Identifying important points (e.g., where the graph intersects the axes)
• Understanding the general shape of the function (increasing, decreasing, asymptotes)
• Using a graphing utility for more precise placement and scaling

In the plotted graph of this function, you will observe that it intersects the x-axis at \( x = 1\) and increases toward positive infinity as x gets larger. Such a graphing exercise not only aids in understanding the algebraic manipulation involved but also offers an intuitive grasp of how logarithmic relationships behave visually.