Problem 59 - College Algebra with Modeling and Visualization

Question

Complete the following. A.Use a table of \(f(x)\) and \(g(x)\) to determine whether \(f(x)=g(x) B.If possible, use properties of logarithms to show that \)f(x)=g(x)$ $$ f(x)=\ln 2 x^{2}-\ln x, \quad g(x)=\ln 2 x $$

Step-by-Step Solution

Answer

Yes, \( f(x) = g(x) \); both simplify to \( \ln(2x) \).

Step 1: Simplify the Expressions

To determine if \( f(x) = g(x) \), let's first simplify each function. \( f(x) = \ln(2x^2) - \ln(x) \) can be rewritten. By using the property \( \ln(a) - \ln(b) = \ln(\frac{a}{b}) \), we get \( \ln\left(\frac{2x^2}{x}\right) = \ln(2x) \). Thus, \( f(x) = \ln(2x) \).

Step 2: Compare the Simplified Expressions

Now that \( f(x) = \ln(2x) \), we compare it with \( g(x) = \ln(2x) \). Since both simplified expressions are \( \ln(2x) \), we conclude that \( f(x) = g(x) \).

Key Concepts

Logarithmic Properties

Understanding logarithmic properties is crucial when dealing with functions like those in our problem. One of the key properties used in the exercise is the quotient rule. This rule states that the difference of two logarithms, such as \( \ln(a) - \ln(b) \), can be rewritten as \( \ln\left(\frac{a}{b}\right) \). This is beneficial because it allows us to simplify complex expressions.

Another important property is the power rule, which states \( \ln(a^b) = b \cdot \ln(a) \). These properties help us break down logarithmic expressions into more manageable forms, making it easier to compare or simplify functions. Mastery of logarithmic properties aids in recognizing these simplifications quickly and efficiently in problems involving logarithms.

Function Simplification

Simplifying functions is a pivotal step in solving equations. In our exercise, \( f(x) = \ln(2x^2) - \ln(x) \), we employed the concept of simplification by combining logarithms.Using the quotient property, we transformed \( f(x) \) into \( \ln(\frac{2x^2}{x}) \).

This expression further simplifies because \( \frac{2x^2}{x} = 2x \), thus leading to \( \ln(2x) \). Simplification helps to reveal the underlying forms of functions that are often disguised by intricate logarithmic operations. It's a process of making complex expressions easier to work with and offers clarity in understanding the true nature of equations.

Mathematical Proof

Mathematical proof is the backbone of confirming hypotheses or comparisons in equations. In our task, we wanted to confirm if \( f(x) \) and \( g(x) \) are indeed equal. Once we simplified both functions to \( \ln(2x) \), we verified their equality by mathematical proof.

The proof involved demonstrating these simplified expressions as one and the same. By applying properties of logarithms accurately, and ensuring all transformations hold true, we effectively used proof to establish the equality of \( f(x) \) and \( g(x) \). Mathematical proofs are essential to validate that solutions meet the conditions set by the original problem and are logically sound.

Function Comparison

Function comparison is a technique used to check if two functions produce the same result for their inputs. In the given problem, both \( f(x) \) and \( g(x) \) were initially different expressions. The task was to determine if they are comparable or equal by simplifying and using mathematical logic.

After simplifying \( f(x) \) to \( \ln(2x) \), we directly compared it with \( g(x) \), which was initially \( \ln(2x) \) without modifications. This comparison showed that the simplified functions were indeed the same. Function comparison helps in understanding not only the equality of functions but also their behavior and similarities. It is fundamental in algebra and calculus, where the goal is to find consistent relational patterns or identities between different mathematical expressions.