Problem 30 - 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.$$f(x)=\log _{2}(x-1)$$.

Step-by-Step Solution

Answer

The graph of the function \(f(x) = \log_2 (x - 1)\) can be produced by using a graphing utility to draw the transformed function \(f(x) = \frac{\ln (x - 1)}{\ln 2}\) with the restriction that \(x > 1\).

Step 1: Identify relevant logarithm formula

The change of base formula for a logarithm is given by \(\log_a b = \frac{\ln b}{\ln a}\) where \( \ln\) denotes the natural logarithm and \(a\) and \(b\) are any two positive numbers. In this exercise, the base \(a\) is 2 and the argument \(b\) is \(x -1\).

Step 2: Substitute into change of base formula

We then substitute \(a = 2\) and \(b = x-1\) into the formula: \(\log_2 (x - 1) = \frac{\ln (x - 1)}{\ln 2}\).

Step 3: Evaluate using a graphing utility

The graphing utility is then used to draw the graph of the function \(f(x) = \frac{\ln (x - 1)}{\ln 2}\). Inputs into this function should be restricted to values of \(x > 1\) to ensure the argument of the logarithm is always positive. The function starts with an undefined value at \(x = 1\) and then increases steadily as \(x\) becomes larger.

Key Concepts

Logarithmic Functions

Logarithmic functions are essential for solving problems that involve exponential growth or decay, among other applications. A logarithmic function is the inverse of an exponential function. It can be expressed as \( f(x) = \log_a(x) \), where \(a\) is the base of the logarithm. This means if \( a^y = x \), then \( \log_a(x) = y \).

Logarithmic functions have several important properties:

• The domain of a logarithmic function is \( x > 0 \) because you can only take the logarithm of positive numbers.
• The range of a logarithmic function is all real numbers.
• Logarithmic functions are continuous and increase steadily, but they never actually touch the x-axis.

The change of base formula allows you to compute logarithms in bases other than 10 or \( e \) (the natural logarithm base). For any positive number \(a eq 1\), \( \log _{a} x\) can be calculated using the formula: \[\log _{a} x= \frac{\ln x}{\ln a}\] This formula is handy for converting between different bases, as it uses the natural logarithm \( \ln \), which is ubiquitous in most scientific calculators and graphing utilities.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a special kind of logarithm with the base \( e \), where \( e \approx 2.71828\). It is widely used in mathematics, physics, and engineering, primarily because of its natural occurrence in calculations involving continuous growth processes.

Key characteristics of the natural logarithm include:

• It shares many properties with other logarithms, such as the laws of logarithms: product, quotient, and power rules.
• Its domain is \( x > 0 \), similar to other logarithmic functions.
• Natural logarithms are used in various natural processes, including calculating compound interest and analyzing radioactive decay.

The natural logarithm is significant in calculus due to its easy differentiation and integration:
- The derivative of \( \ln(x) \) is \( \frac{1}{x} \), which is highly useful in applying calculus to real-world problems.- The integral of \( \ln(x) \) produces a formula that involves the logarithm itself, making it useful in solving complex integrals.

Graphing Utility

A graphing utility is an invaluable tool in visualizing mathematical functions and their properties. It allows you to input equations and understand their behavior graphically. When working with the function \(f(x) = \log_2(x-1)\), a graphing utility can help in illustrating how the function behaves, especially at points where traditional calculations may be difficult.

When using a graphing utility:

• First, ensure the equation is input correctly, converting it if necessary using the change of base formula. For \( \log_2(x-1) \), use \( \frac{\ln(x-1)}{\ln(2)} \).
• Set the domain restrictions accurately. Since the argument \( x-1 \) must be positive, \( x > 1 \) is essential.
• Observe key features such as intercepts, asymptotes, and end behavior. For \( \log_2(x-1) \), note the behavior near \( x=1 \) and as \( x \) extends towards infinity.

Graphing utilities simplify these tasks by providing visual feedback, helping in understanding and solving complex functions more intuitively.