Problem 95 - College Algebra

Question

In Exercises \(89-102,\) determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. $$ \ln (5 x)+\ln 1=\ln (5 x) $$

Step-by-Step Solution

Answer

The equation \(\ln (5x) + \ln 1 = \ln (5x)\) is true.

Step 1: Simplify the equation

Given the equation, \(\ln (5x) + \ln 1 = \ln (5x)\), the logarithm of 1 in any base is zero. Thus, this equation simplifies to \(\ln (5x) + 0 = \ln (5x)\). This means \(\ln (5x) = \ln (5x)\).

Step 2: Analyze the equality

After the simplification in step 1, it is clear that both sides of the equation are equal, as they are essentially stating the same thing.

Step 3: State the conclusion

Since the equation holds true after its simplification, we can then conclude that the given equation is true.

Key Concepts

Logarithmic Properties

Understanding the properties of logarithms is essential in solving many algebraic equations. One important property is that the logarithm of 1, regardless of the base, always equals 0. This is helpful because when we encounter expressions like \( \ln(1) \), we immediately know it simplifies to 0. Similarly, the identity property of logarithms states that \( \ln(a) + \ln(b) = \ln(ab) \). These properties allow us to simplify complex logarithmic equations, making them easier to manage and solve.

Equation Simplification

Simplifying equations is a critical step in solving or proving mathematical statements. In our original exercise, we start with the equation \( \ln(5x) + \ln(1) = \ln(5x) \). Applying logarithmic properties, we recognize that \( \ln(1) \) is 0. This transforms the equation into \( \ln(5x) + 0 = \ln(5x) \). Simplification helps clarify the equation, enabling us to better understand its structure and whether both sides can be accurately equated or further analyzed.

Mathematical Proof

A mathematical proof involves logically establishing the truth or falsehood of a statement. Proofs are structured around known principles and properties. In the exercise, the proof involves demonstrating that \( \ln(5x) = \ln(5x) \) is indeed a true statement—supported by the prior simplification. Proofs typically begin with assumptions or given information, followed by a series of logical deductions using mathematical properties (like those of logarithms) to arrive at a definitive conclusion. Understanding proofs is essential for verifying results rigorously.

Equality Verification

Equality verification in mathematics ensures that two expressions are truly equal. In algebraic equations, this process starts by simplifying each side separately to reach a common form. In the problem presented, both sides of the equation, after simplification, were \( \ln(5x) \). Confirming this equality was straightforward because both expressions shared the same format and value. Equality verification is crucial for validating mathematical equations, ensuring they hold under specific conditions or transformations, and are logically consistent.