Problem 70 - Precalculus : Building Concepts and Connections

Question

Find the domain of each function. Use your answer to help you graph the function, and label all asymptotes. $$g(s)=\log _{1 / 2} s$$

Step-by-Step Solution

Answer

The domain of the function \(g(s)=\log _{1 / 2} s\) is \(s>0\) and the vertical asymptote is \(s=0\). The graph of the function decreases from left to right with the vertical asymptote drawn at \(s=0\).

Step 1: Determine the Domain

A function's domain encompass all possible input values (here: s). For all base logarithms \(b\), whether \(b > 1\) or \(00\).

Step 2: Identify asymptotes

In logarithmic functions, horizontal asymptotes do not exist. However, vertical asymptotes can be found where the function is undefined. As the domain is \(s>0\), it approaches zero but never reaches it, therefore making \(s=0\) a vertical asymptote.

Step 3: Graph the function and label asymptotes

When graphing, depict the function \(g(s)=\log _{1 / 2} s\) as it appears in the cartesian coordinate-system. Draw the vertical asymptote at \(s=0\). In this case where \(b<1\), the graph will be decreasing from left to right across the x-axis.

Key Concepts

Graphing Logarithmic Functions

To graph a logarithmic function, such as \( g(s) = \log_{1/2}(s) \), it's essential to understand its behavior and characteristics. Since logarithmic functions are the inverses of exponential functions, their graphs reflect a specific set of patterns. For functions with a base smaller than 1, like our example where \( b = 1/2 \), the graph is a curve that decreases as it moves from left to right, never crossing the y-axis (the line \( s = 0 \)).

To start graphing, plot points by choosing positive input values for \( s \) and compute the corresponding \( g(s) \) values. It might be helpful to use a table of values to get accurate points on the graph. As you plot more points, you'll see the curve emerge, getting closer and closer to the y-axis as \( s \) approaches zero, but never actually touching it.

Remember that logarithmic scales differ from linear ones; distances on the graph are not uniform as they represent the power to which the base must be raised to achieve the result. This is why logarithmic functions are particularly useful in depicting data that spans several orders of magnitude, like the Richter scale used to measure earthquakes.

Identifying Asymptotes

An asymptote is a line that a function approaches but never actually reaches or crosses. In the context of logarithmic functions, we are typically talking about vertical asymptotes. For the function \( g(s) = \log_{1/2} s \), the vertical asymptote is the y-axis or where \( s = 0 \).

Imagine drawing a line parallel to the x-axis right up against the y-axis without touching it; this represents the line s=0, the vertical asymptote. The function will never intersect with this line, no matter how far the graph is extended. As you graph the function, represent the asymptote with a dashed line to indicate that the function's values approach infinity without reaching the line.

Understanding where the asymptotes are located helps in sketching the overall shape of the graph. In some cases, there may also be horizontal asymptotes to consider, but this does not apply to logarithmic functions, as they do not level off as x approaches positive or negative infinity.

Logarithm Properties

The logarithm function has unique properties that can simplify solving equations and transforming expressions. Here are some of the fundamental properties of logarithms:

• Product Property: \( \log_b(mn) = \log_b(m) + \log_b(n) \), meaning the logarithm of a product is the sum of the logarithms.
• Quotient Property: \( \log_b(\frac{m}{n}) = \log_b(m) - \log_b(n) \), which tells us the logarithm of a quotient is the difference between the logarithms.
• Power Property: \( \log_b(m^n) = n \cdot \log_b(m) \), indicating that the logarithm of a power is the exponent times the logarithm of the base.
• Change of Base Formula: \( \log_b(m) = \frac{\log_k(m)}{\log_k(b)} \), where you can convert a logarithm to a different base \( k \) if necessary, which is particularly useful when using calculators.

These properties are essential tools when working with logarithms, as they allow for the manipulation of terms to either combine or separate logarithmic expressions. This makes it easier to solve complex logarithmic equations and can be particularly helpful when dealing with variable expressions within logarithms.