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Chapter 3: Problem 91

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. Extraneous solutions arise when the process of solving leads to results that don't satisfy the necessary conditions of a logarithm.

Step by step solution

01

Understand the concept of an extraneous solution

An extraneous solution is a result acquired during the process of solving an equation, which is not a genuine solution. In simple terms, if substituting the result back into the original equation makes the equation invalid, then the result is an extraneous solution.
02

Recognize the context of logarithmic equations

In a logarithmic equation such as \( \log_b{A} = x \), it is assumed the base 'b' is greater than 0 and not equal to 1, and the argument 'A' is a positive real number. Extraneous solutions may arise when we violate these conditions during the process of solving the equation.
03

Explain the possibility of multiple extraneous solutions

It is indeed possible for a logarithmic equation to have more than one extraneous solution. This may occur if there are multiple logarithmic terms, variables, or quantities that do not satisfy the basic logarithm rules.

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Most popular questions from this chapter

Condensing a Logarithmic Expression In Exercises \(67-82,\) condense the expression to the logarithm of a single quantity. 2$$[3 \ln x-\ln (x+1)-\ln (x-1)]$$

True or False? In Exercises \(97-102,\) determine whether the statement is true or false given that \(f(x)=\ln x .\) Justify your answer. $$f(a x)=f(a)+f(x), \quad a>0, \quad x>0$$

In Exercises \(85-88,\) use the following information. The relationship between the number of decibels \(\beta\) and the intensity of a sound I in watts per square meter is given by $$ \boldsymbol{\beta}=10 \log \left(\frac{I}{10^{-12}}\right) $$ Use the properties of logarithms to write the formula in simpler form, and determine the number of decibels of a sound with an intensity of \(10^{-6}\) watt per square meter.

Rewriting a Logarithm In Exercises \(7-10\) , rewrite the logarithm as a ratio of (a) common logarithms and (b) natural logarithms. $$\log _{2.6} x$$

Using the Change-of-Base Formula In Exercises \(11-14,\) evaluate the logarithm using the change-of-base formula. Round your result to three decimal places. $$\log _{3} 0.015$$
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