Problem 1 - Intermediate Algebra

Question

Fill in the blanks. \(f(x)=\log _{2} x\) and \(g(x)=\log x\) are examples of _____ functions.

Step-by-Step Solution

Answer

Logarithmic

Step 1: Identify the Type of Function

The first function, \(f(x) = \log_{2} x\), is a logarithm with base 2, while \(g(x) = \log x\) is a logarithm with base 10 (commonly known as the common logarithm). Both functions involve logarithms.

Step 2: Define the Function Category

Logarithmic functions are generally characterized by the expression \(f(x) = \log_{b} x\), where \(b\) represents the base of the logarithm. Thus, both functions provided are of this form.

Step 3: Fill in the Blank with the Correct Term

Since \(f(x)\) and \(g(x)\) are characterized by having logarithms as their central operations, each function is a logarithmic function. Therefore, the blank should be filled with the term "logarithmic."

Key Concepts

Logarithm

Logarithms are fascinating mathematical tools that help us understand the concept of scaling and growth in a multiplicative way. A logarithm essentially tells us how many times a certain number, known as the base, has to be multiplied with itself to reach another number. For instance, if we have the expression \(\log_{2} 8\), this is asking "to what power must 2 be raised to produce 8?" The answer would be 3, because \(2^3 = 8\). This is what makes logarithms particularly useful in a variety of scientific and engineering applications.
The logarithmic function \(f(x) = \log_{b} x\) shows the relationship between the exponent and the result of raising the base \(b\) to that exponent. Key properties of logarithms include:

• Product rule: \(\log_{b}(xy) = \log_{b}x + \log_{b}y\)
• Quotient rule: \(\log_{b}\left(\frac{x}{y}\right) = \log_{b}x - \log_{b}y\)
• Power rule: \(\log_{b}(x^k) = k \cdot \log_{b}x\)

These properties make computing complex expressions more manageable and are frequently encountered in calculus and higher mathematics.

Common Logarithm

The common logarithm refers to a logarithm with a base of 10, denoted as \(\log x\). When no base is written, it's typically assumed to be 10. This simplifies calculations as we are often dealing with base-10 numbers in our everyday numerical system.
In scientific calculations, common logarithms are especially useful. For instance, they are used to calculate the pH level in chemistry, which measures the acidity of a solution. The pH is determined as the common logarithm of the concentration of hydrogen ions. The formula for pH is \(\text{pH} = -\log[H^+]\), demonstrating the practicality of this base-10 log.
One notable advantage of the common logarithm is its integration with calculators and logarithmic tables, making it a handy tool for quick computations. Additionally, it fits neatly into our decimal number system, allowing for seamless calculations when dealing with scalable measurements like decibels in audio engineering.

Function Categories

Functions are categorized based on their operational characteristics and expressions. In mathematics, a function can fall under various types such as linear, quadratic, polynomial, and logarithmic functions, among others. Logarithmic functions, in particular, are characterized by the use of logarithms within their expressions.
For example, both \(f(x) = \log_{2} x\) and \(g(x) = \log x\) fit into the logarithmic function category because they are centered around the operation of taking a logarithm of a variable, \(x\). Logarithmic functions are vital in contexts where exponential growth or decay is involved.
Understanding the category of a function helps in selecting appropriate methods for solving equations and graphing. In logarithmic functions, the inverse relationship to exponential functions is particularly noteworthy, as it provides essential insights into the behavior of equations and their real-world applications. This categorization thus plays a critical role in professional fields like data science, economics, and engineering.