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Chapter 3: Problem 143

Is it possible for a logarithmic equation to have more than one extraneous solution? Explain.

Short Answer

Expert verified
Yes, a logarithmic equation can have more than one extraneous solution, typically resulting from the mathematical manipulations performed during the solution process.

Step by step solution

01

Define Logarithmic Equation and Extraneous Solution

A logarithmic equation is an equation that involves the logarithm of a variable containing expression. Extraneous solutions are those solutions that are not actually valid for the original equation, but they arise due to the mathematical manipulation during the solution process.
02

Potential for Multiple Extraneous Solutions

The answer to whether a logarithmic equation can have more than one extraneous solution is affirmative, yes. When solving logarithmic equations, the properties of logarithms are often used to manipulate the equation and isolate the variable. This manipulation often includes operations that could produce extraneous solutions, such as squaring both sides of an equation. When these equations are later tested for validity, multiple extraneous solutions could potentially be identified.
03

Reiteration and Understanding

So, in conclusion, extraneous solutions are generally induced by the process of solving the equation itself. In the case of logarithmic equations, mathematical transformations, like squaring or cubing, that are employed to solve, can give rise to more than one extraneous solution. The presence of multiple extraneous solutions doesn't necessarily mean there's an error in the math; rather, it's just a byproduct of the necessary steps for finding the true, valid solution(s).

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