Problem 104 - Precalculus with Limits

Question

Using the Change-of-Base Formula In Exercises \(103-106,\) use the change-of-base formula to rewrite the logarithm as a ratio of logarithms. Then use a graphing utility to graph the ratio. $$f(x)=\log _{1 / 2} x$$

Step-by-Step Solution

Answer

The original logarithmic function \(f(x) = \log _{1 / 2} x\) can be rewritten using the change-of-base formula as \(f(x) = -ln(x)/ln(2)\) and graphed to show a rapid decline as x increases.

Step 1: Apply the Change-of-Base Formula

To rewrite the given logarithmic function using the change-of-base formula, choose a new base. A common choice is base 10 or base e. The change-of-base formula is \(log_b(a) = log_c(a)/log_c(b)\), where c can be any positive number except 1. Here, we will transform the original logarithm to the natural base (e). So the function becomes: \(f(x) = ln(x)/ln(1/2)\).

Step 2: Simplify the Expression

The function \(f(x) = ln(x)/ln(1/2)\) can be further simplified. Remember that \(ln(1/2) = -ln(2)\), because 1/2 is the reciprocal of 2, and the logarithm of a reciprocal is the negative of the logarithm. This allows us to simplify the function as: \(f(x) = -ln(x)/ln(2)\).

Step 3: Graph the Function

At this point, the function can be graphed using a graphing tool. Any standard graphing software can be used. The graph will show that the function \(f(x) = -ln(x)/ln(2)\) takes on the shape of a logarithmic decay, continuing to steeply decline as x increases.

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and are widely used in various fields, including mathematics, science, and engineering. At the heart of understanding a logarithmic function is knowing that it corresponds to the question: 'To what exponent must we raise a base to get a certain number?' In mathematical terms, if we have a function defined as f(x) = log_b(x), it’s asking, 'What power should we raise b to get x?'.

The change-of-base formula is a powerful tool when working with logarithms. It allows us to convert a logarithm with any base to a logarithm of our choosing, which is particularly useful when we want to simplify calculations or when using graphing utilities that only support certain bases, such as base 10 or the natural base e. Simplifying expressions using the change-of-base formula can also reveal the underlying behavior of the function, such as its growth or decay rate.

Natural Logarithm (ln)

The natural logarithm, denoted as ln, uses the irrational number e (approximately 2.718) as its base. It's a constant that arises naturally in mathematics when dealing with continuous growth or decay processes. For example, ln(x) answers the question, 'To what power do we raise e to obtain the number x?'.

Using the natural logarithm simplifies many aspects of calculus and differential equations because the derivative of ln(x) with respect to x is 1/x, which makes it advantageous to work with compared to other bases. When we apply the change-of-base formula in the exercise, it becomes evident that ln is a convenient choice, allowing further simplification of the expression. Understanding the properties of ln, such as ln(1/2) being the negative of ln(2), underlines the versatility of natural logarithms and the elegance they bring to complex mathematical expressions.

Graphing Logarithmic Functions

Graphing logarithmic functions involves understanding their characteristic shape, which typically shows a rapid increase or decrease and then levels off. For a basic logarithmic function f(x) = log_b(x), the graph will pass through the point (1, 0), since any base raised to the power of 0 will equal 1. It will also approach the y-axis asymptotically since logarithms are undefined for 0 and negative numbers.

The graph of a logarithmic function can reflect growth or decay, which depends on the base of the logarithm. If the base is greater than 1, the function shows growth; if the base is between 0 and 1, as in the exercise with base 1/2, the function shows decay. By using a graphing utility and the simplifications made through the change-of-base formula, students can visualize the rate and direction of change, adding depth to their understanding of logarithmic behavior in varying conditions.