Problem 34 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the change-of-base formula \(\log _{a} x=(\ln x) /(\ln a)\) and a graphing utility to graph the function.$$f(x)=\log _{3} \sqrt{x}$$.

Step-by-Step Solution

Answer

The transformed function is \(f(x) = \frac{1}{2} \frac{ln x}{ln 3}\). This can be graphed using a graphing utility.

Step 1: Simplify the square root

Since \(\sqrt{x}\) is equivalent to \(x^{1/2}\), the function can be rewritten as: \(f(x)=\log _{3} x^{1/2}\)

Step 2: Apply change-of-base formula

Using the change-of-base formula, the logarithm can be rewritten as: \(f(x) = \frac{\ln x^{1/2}}{\ln 3}\).

Step 3: Simplify the numerator

According to the laws of logarithms, \(\ln x^{1/2} = \frac{1}{2} ln x\). This simplifies our function to: \(f(x) = \frac{1}{2} \frac{ln x}{ln 3}\)

Key Concepts

Change-of-Base Formula

The change-of-base formula is a handy tool when working with logarithmic expressions, especially when you're dealing with a base that's not commonly used. This formula is given by:\[\log_{a} x = \frac{\ln x}{\ln a}\]This allows you to convert logarithms with any base \(a\) into natural logarithms (base \(e\)), which are often much easier to handle, especially when using calculators or graphing utilities that primarily operate with natural logarithms. In our example, \(f(x) = \log_{3} \sqrt{x}\), to graph this function, we will first apply the change-of-base formula.

• The expression \(\log_{3} \sqrt{x}\) is converted using natural logarithms: \(f(x) = \frac{\ln x^{1/2}}{\ln 3}\).
• This rewrite also helps simplify the calculations or when evaluating the function for specific values of \(x\) using a graphing calculator or software.

Using this formula makes non-standard base logarithmic functions easier to manage computationally.

Laws of Logarithms

The laws of logarithms are essential rules that help us manipulate and simplify logarithmic expressions. They are valuable tools in algebra and calculus to solve logarithmic equations or set up expressions for graphing. In our function, \(f(x) = \log_{3} \sqrt{x}\), applying the laws of logarithms is a crucial step.To simplify \(\ln x^{1/2}\), we use one of these laws that states:\[\ln (x^r) = r \cdot \ln x\]Applying this to \(x^{1/2}\), we get:\[\ln x^{1/2} = \frac{1}{2} \ln x\]By understanding and applying these rules, we rewrite our function as:\[f(x) = \frac{1}{2} \cdot \frac{\ln x}{\ln 3}\]

• This shows how the exponent affects the logarithm calculation and how constants can be factored out for simplification.
• Recognizing these transformations is key to working efficiently with log functions.

These logarithmic laws make it simpler to perform further evaluations, whether by hand or using digital tools.

Graphing Utility

Graphing utilities are invaluable when visualizing complex functions, such as logarithmic expressions with different bases. Once an expression is rewritten using the change-of-base formula, it becomes much easier to graph. Our example, \(f(x) = \frac{1}{2} \cdot \frac{\ln x}{\ln 3}\), is ready for this step.Here's how you might approach graphing this function:

• Enter the equation \(f(x) = \frac{1}{2} \cdot \frac{\ln x}{\ln 3}\) into a graphing calculator or software.
• Ensure that the calculator is set to interpret natural logarithms correctly (which it should do by default).
• Adjust the window settings to view the full behavior of the function over the domain of interest. For example, because the domain of \(\ln x\) is \(x > 0\), set your x-range accordingly.
• Observe any key characteristics such as asymptotes, intersections, and overall shape.

Graphing utilities provide a visual insight that complements the analytical approach, helping to better understand the nature and behavior of the function across its domain.