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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, it is possible for a logarithmic equation to have more than one extraneous solution. This occurs when multiple solutions of the manipulated equation do not satisfy the condition of the original logarithmic equation.

Step by step solution

01

Understand Extraneous Solutions

An extraneous solution is a solution that emerges from the process of solving the equation, but does not satisfy the original equation. For instance, in the process of solving an equation, it might require to square both sides, which produces additional solutions that weren't solutions to the original equation.
02

Know the Properties of Logarithmic Equations

Logarithmic equations typically have restrictions on their solutions - mainly due to the fact that the logarithms of negative numbers and zero are undefined. So, solutions suggesting log of a negative number or zero are mathematically invalid and hence, are termed extraneous.
03

Apply the Concepts

Given that it's possible to obtain multiple solutions when solving logarithmic equations (not forgetting the fact that logarithmic equations can contain more than one logarithmic term), and that not all of these solutions might satisfy the original equation due to the restrictions of logarithmic functions, it is indeed possible for a logarithmic equation to have more than one extraneous solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Logarithmic Equations
Logarithmic equations involve the logarithm of a variable or number. A basic form looks like \( \log_b(x) = y \), indicating that \( b^y = x \). Solving logarithmic equations often requires your understanding of converting between logarithmic and exponential forms.

In equations, we cross-reference logarithmic expressions to find variables. This involves steps like:
  • Aligning terms with the same base.
  • Using properties of logarithms to simplify (which we'll discuss later).
  • Restructuring the equation into exponential form to isolate the variable.
When performing these operations, extra caution is needed to confirm that the derived solutions satisfy the original equation. This leads us to the next concept.
Exploring Solution Restrictions
Solution restrictions in logarithms stem from their properties. Logarithms of negative numbers and zero do not exist in the real number set. This naturally imposes boundaries on the potential solutions to logarithmic equations.

When solving an equation, let鈥檚 say \( \log_2(x) = 3 \), we immediately know that \( x = 8 \). Here it鈥檚 evident that \( x \) cannot be zero or negative. But complications arise when transformations, like squaring both sides during transformations, introduce solutions.
  • These solutions need verification because they might not reflect valid outputs for original logarithmic constraints.
  • Sometimes while processing the equation, multiple solutions emerge 鈥� some of which might not satisfy the original constraints.
This leads us to extraneous solutions.
Utilizing Properties of Logarithms
The properties of logarithms are powerful tools. Understanding these properties helps to simplify logarithmic equations appropriately. Here are some key properties:
  • Product Property: \( \log_b(MN) = \log_b(M) + \log_b(N) \)
  • Quotient Property: \( \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) \)
  • Power Property: \( \log_b(M^p) = p\cdot\log_b(M) \)
These help in breaking down complex expressions into manageable parts.

When navigating through logarithmic equations, carefully applying these properties post-manipulation is crucial. They not only help in simplifying but also ensuring solutions found in manipulations adhere to the original equations, thus revealing any extraneous solutions.

In summary, proper use of logarithmic properties combined with understanding solution restrictions ensures accurate solutions to logarithmic equations.

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Most popular questions from this chapter

Approximate the logarithm using the properties of logarithms, given \(\log _{b} 2 \approx 0.3562, \log _{b} 3 \approx 0.5646,\) and \(\log _{b} 5 \approx 0.8271.\) $$\log _{b} \sqrt{2}$$

Condense the expression to the logarithm of a single quantity. $$\frac{1}{3}\left[2 \ln (x+3)+\ln x-\ln \left(x^{2}-1\right)\right]$$

Condense the expression to the logarithm of a single quantity. $$-4 \log _{6} 2 x$$

Use the properties of logarithms to expand the expression as a sum, difference, and or constant multiple of logarithms. (Assume all variables are positive.) $$\log _{10} \frac{x y^{4}}{z^{5}}$$

The approximate lengths and diameters (in inches) of common nails are shown in the table. Find a logarithmic equation that relates the diameter \(y\) of a common nail to its length \(x\). $$\begin{array}{|c|c|}\hline \text { Length, } x & \text { Diameter, } y \\\\\hline 1 & 0.072 \\\\\hline 2 & 0.120 \\\\\hline 3 & 0.148 \\ \hline 4 & 0.203 \\\\\hline 5 & 0.238 \\\\\hline\end{array}$$
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