Problem 31 - 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.$$f(x)=\log _{1 / 2}(x-2)$$ .

Step-by-Step Solution

Answer

The graph of the function \(f(x)=\log _{1 / 2}(x-2)\) is the same shape as a standard log function, but mirrored in the y-axis because of the negative sign and shifted to the right by 2 units due to the transformation \(x-2\).

Step 1: Apply the Change of Base Formula

Start by applying the change of base formula to the function \(f(x)=\log _{1 / 2}(x-2)\). Using the formula, we turn the base \(1/2\) log into a natural logarithm. The change of base formula is \(\log_{a} x = (\ln x)/(\ln a)\).Applying the formula gives us a transformed function:\[f(x)= \frac{\ln(x - 2)}{\ln(1 / 2)}.\]

Step 2: Simplify the Expression

Simplify the expression to make it easier to graph. We know that \(\ln (1/2)\) is a negative number because \(1/2\) is less than 1. So we can rewrite the expression as:\[f(x)=- \frac{\ln(x - 2)}{\ln 2}.\]

Step 3: Graph the Function using a Graphing Utility

With the function simplified, we can now graph it using a graphing utility. Plot \(x\) on the x-axis and \(f(x)\) on the y-axis. Since this is a logarithmic function, remember that the log function is undefined for \(x ≤ 0\). So, make sure to start your graph at \(x > 2.0\). Plot several x and y pairs so that the general shape of the function can be seen. This will give the graph of the function \(f(x)=- \frac{\ln(x - 2)}{\ln 2}\).

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions, and they play a crucial role in various mathematical and real-world applications, like measuring sound intensity or the acidity of a solution. The general form of a logarithmic function is \(y = \log_b(x)\), where \(b\) is the base and \(x\) is the argument of the logarithm. The function represents the power to which the base \(b\) must be raised to produce the number \(x\).

It's important to remember that the domain of a logarithmic function is \(x > 0\), as you cannot take the logarithm of zero or a negative number. Moreover, a base of a logarithm must be positive and not equal to 1, because these values would not fulfill the purpose of a logarithmic function, which is to map the multiplicative scale of growth onto an additive scale.

Natural Logarithm

The natural logarithm, represented as \(\ln x\), is a special type of logarithm with the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. This base is chosen because it yields simpler derivatives and integrals, making it essential in calculus and its applications.

The natural logarithm has properties that make it useful for simplifying complex mathematical expressions. For example, \(\ln(1) = 0\) and \(\ln(e) = 1\). The function grows slowly; for large values of \(x\), the function increases at a rate inversely proportional to \(x\), which makes it a useful tool in modeling processes that grow or decay gradually over time.

Graphing Logarithmic Function

Graphing a logarithmic function like \(f(x) = \log_{1/2}(x - 2)\) involves understanding the behavior and characteristics of logarithms. To graph this function using a graphing utility, you should recognize some key features:

• The graph has a vertical asymptote at \(x = 2\), to the left of which the function is undefined.
• Since \(1/2 < 1\), the function is decreasing, with a slope that gets less steep as \(x\) increases.
• The y-values will get more negative as \(x\) increases.

Before plotting points on the graphing utility, use transformations of the function to get a more straightforward equation by applying the change of base formula. This method helps in getting a clearer understanding of how to graph the logarithmic function by viewing it through the lens of a natural log, which is often easier to work with and graph on standard calculators and graphing tools.