Problem 78 - Precalculus with Calculus Previews

Question

For \(f(x)=\log _{b} x\), show that \(f\left(x_{1} x_{2}\right)=f\left(x_{1}\right)+f\left(x_{2}\right) .\)

Step-by-Step Solution

Answer

The property \(f(x_1x_2) = f(x_1) + f(x_2)\) is true for logarithm functions.

Step 1: Understand the Logarithmic Property

The property of logarithms we are going to use is: The logarithm of a product is equal to the sum of the logarithms of the factors.Mathematically, it is stated as:\[\log_b (x_1 \cdot x_2) = \log_b (x_1) + \log_b (x_2)\]

Step 2: Apply the Property to f(x) Function

Given the function \(f(x) = \log_b x\), we apply the logarithmic property to \(f(x_1x_2)\):\[f(x_1x_2) = \log_b (x_1x_2)\]Using the logarithmic property from Step 1, this becomes:\[\log_b(x_1) + \log_b(x_2)\]

Step 3: Relate to Function Values

Substituting back to the function definitions, we get:\[f(x_1x_2) = f(x_1) + f(x_2)\]This shows that the function evaluated at the product of two arguments is the same as the sum of the function values of each individual argument.

Key Concepts

Logarithmic Functions

In mathematics, a logarithmic function is one of the fundamental functions that has a wide range of applications in various fields. Logarithms are the inverse operations of exponentials. This means that they allow us to find the exponent that a base number is raised to, to get a specific value.
For example, if we have \( b^y = x \), then \( \log_b x = y \).
Logarithmic functions are essential when dealing with exponential growth or decay, as they offer a way to linearize data that multiplies or divides by a constant rate.

• **Base of the Logarithm:** This is the number that is consistently raised to a power to get another number. In \( \log_b x \), \( b \) is the base.
• **Natural Logarithm:** A specific type of logarithm where the base is Euler’s number ( e ≈ 2.718), denoted as \( \ln(x) \).
• **Common Logarithm:** When the base is 10, the logarithm is called a common logarithm, denoted as \( \log(x) \).

Understanding these functions is key to solving problems that involve scaling operations, such as sound intensity, pH in chemistry, and data analysis in statistics.

Product Property of Logarithms

The product property of logarithms is an essential tool that simplifies the process of handling multiplication within logarithmic expressions. According to this property, the logarithm of a product is equal to the sum of the logarithms of its factors.
This can be expressed mathematically as:
\[\log_b (x_1 \cdot x_2) = \log_b (x_1) + \log_b (x_2)\]Where \( b \) is the base of the logarithms, and \( x_1 \) and \( x_2 \) are the values of interest.

• This property is frequently used in simplifying expressions and solving exponential equations.
• It is particularly useful when multiplying large numbers, as it reduces the operation to addition.
• This property stems from the exponent rules, where multiplication of terms with the same base equals adding the exponents.

Understanding and applying these properties allow for more effective and efficient problem-solving in precalculus and beyond.

Precalculus

Precalculus is an intermediate course of study that lays the groundwork for more advanced mathematics courses such as calculus. It encompasses a broad range of concepts designed to prepare students for dealing with complex mathematical operations.
Some of the fundamental components of precalculus include:

• **Functions and Graphs:** Understanding the behavior of different types of functions, including polynomial, rational, logarithmic, and exponential functions.
• **Algebraic Manipulations:** Skills in algebra are crucial, as they form the basis for manipulating equations and expressions used across different topics.
• **Trigonometry:** The study of triangles involves learning about angles, sine, cosine, and tangent, which are vital in many applications of calculus.

Precalculus also involves the study of sequences and series and introduces students to limits and continuity—concepts that are vital in calculus.
By mastering precalculus, students are well-prepared to transition smoothly into calculus, where they will apply these principles to rate of change and accumulation problems.