Extraneous Solutions In Logarithmic Equations A Detailed Guide

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In the realm of mathematics, particularly when dealing with logarithmic equations, the concept of extraneous solutions often emerges as a critical aspect to understand. These solutions, while seemingly valid through the algebraic manipulation of an equation, do not satisfy the original equation's conditions. This article delves into the intricacies of extraneous solutions within the context of logarithmic equations, using a specific example to illustrate the process of identifying and excluding them.

The given logarithmic equation is:

log⁡3(18x3)−log⁡3(2x)=log⁡3144\log _3\left(18 x^3\right)-\log _3(2 x)=\log _3 144

To solve this equation, we will meticulously walk through each step, highlighting potential pitfalls that can lead to extraneous solutions. The goal is not only to find the solutions but also to understand why certain solutions might be invalid.

Step-by-Step Solution

1. Applying Logarithmic Properties

The first step in solving the equation involves using the properties of logarithms to simplify the expression. Specifically, we will use the quotient rule, which states that the logarithm of the quotient of two numbers is the difference of their logarithms. Mathematically, this is expressed as:

log⁡b(m)−log⁡b(n)=log⁡b(mn)\log_b(m) - \log_b(n) = \log_b(\frac{m}{n})

Applying this rule to our equation, we get:

log⁡3(18x32x)=log⁡3144\log _3\left(\frac{18 x^3}{2 x}\right)=\log _3 144

2. Simplifying the Argument

Next, we simplify the argument inside the logarithm:

18x32x=9x2\frac{18 x^3}{2 x} = 9x^2

Thus, our equation becomes:

log⁡3(9x2)=log⁡3144\log _3(9x^2) = \log _3 144

3. Equating the Arguments

Since the logarithms on both sides of the equation have the same base, we can equate their arguments:

9x2=1449x^2 = 144

4. Solving for x

Now, we solve the quadratic equation for x. Divide both sides by 9:

x2=1449=16x^2 = \frac{144}{9} = 16

Taking the square root of both sides, we get:

x=±16=±4x = \pm \sqrt{16} = \pm 4

So, the potential solutions are x = 4 and x = -4.

5. Checking for Extraneous Solutions

This is the crucial step where we determine whether our solutions are valid or extraneous. Extraneous solutions arise because logarithms are only defined for positive arguments. Therefore, we must check each potential solution in the original equation to ensure that it does not result in taking the logarithm of a non-positive number.

a. Checking x = 4

Substitute x = 4 into the original equation:

log⁡3(18(4)3)−log⁡3(2(4))=log⁡3144\log _3(18(4)^3) - \log _3(2(4)) = \log _3 144

log⁡3(18⋅64)−log⁡3(8)=log⁡3144\log _3(18 \cdot 64) - \log _3(8) = \log _3 144

log⁡3(1152)−log⁡3(8)=log⁡3144\log _3(1152) - \log _3(8) = \log _3 144

Using the quotient rule in reverse:

log⁡3(11528)=log⁡3144\log _3\left(\frac{1152}{8}\right) = \log _3 144

log⁡3(144)=log⁡3144\log _3(144) = \log _3 144

Since this holds true, x = 4 is a valid solution.

b. Checking x = -4

Substitute x = -4 into the original equation:

log⁡3(18(−4)3)−log⁡3(2(−4))=log⁡3144\log _3(18(-4)^3) - \log _3(2(-4)) = \log _3 144

log⁡3(18⋅(−64))−log⁡3(−8)=log⁡3144\log _3(18 \cdot (-64)) - \log _3(-8) = \log _3 144

log⁡3(−1152)−log⁡3(−8)=log⁡3144\log _3(-1152) - \log _3(-8) = \log _3 144

Here, we encounter logarithms of negative numbers, which are undefined in the real number system. Therefore, x = -4 is an extraneous solution.

6. Conclusion

From our analysis, x = 4 is a valid solution, while x = -4 is an extraneous solution because it results in taking the logarithm of a negative number. Therefore, the extraneous solution to the given logarithmic equation is x = -4.

Understanding Logarithmic Functions

At the heart of solving logarithmic equations lies a thorough understanding of logarithmic functions. A logarithmic function is the inverse of an exponential function. The logarithm of a number x with respect to a base b is the exponent to which b must be raised to produce x. Mathematically, this is written as:

y=log⁡b(x)y = \log_b(x)

where b is the base, and x is the argument. The base b must be a positive number not equal to 1, and the argument x must be positive.

The domain of a logarithmic function is the set of all positive real numbers. This is because the logarithm of a non-positive number is undefined. This constraint is the primary reason why extraneous solutions can arise in logarithmic equations. When solving these equations, algebraic manipulations might lead to solutions that, when plugged back into the original equation, result in taking the logarithm of a non-positive number.

Common Mistakes and How to Avoid Them

Solving logarithmic equations can be challenging, and there are several common mistakes that students make. Awareness of these pitfalls can help in avoiding them.

1. Forgetting to Check for Extraneous Solutions

As illustrated in our example, this is perhaps the most critical step. Always substitute the solutions back into the original equation to ensure they are valid.

2. Incorrectly Applying Logarithmic Properties

Logarithmic properties are powerful tools for simplifying equations, but they must be applied correctly. For instance, the quotient rule, product rule, and power rule have specific conditions under which they are valid. Misapplication can lead to incorrect solutions.

3. Ignoring the Domain of Logarithmic Functions

Remember that the argument of a logarithm must be positive. Any solution that results in a non-positive argument is extraneous.

4. Algebraic Errors

Careless algebraic manipulations can lead to incorrect solutions. Double-check each step, especially when dealing with quadratic equations or fractions.

Real-World Applications of Logarithmic Equations

Logarithmic equations are not just abstract mathematical concepts; they have numerous applications in various fields. Understanding these applications can provide a deeper appreciation for the importance of mastering logarithmic equations.

1. Finance

In finance, logarithmic scales are used to represent financial data, such as stock prices, over long periods. Logarithmic equations are also used in calculating compound interest and loan amortization.

2. Science and Engineering

  • Chemistry: The pH scale, which measures the acidity or alkalinity of a solution, is based on logarithms. The pH is defined as the negative logarithm of the hydrogen ion concentration.
  • Physics: Logarithmic scales are used to measure sound intensity (decibels) and earthquake magnitude (Richter scale).
  • Engineering: Logarithmic functions are used in signal processing, control systems, and data compression.

3. Computer Science

In computer science, logarithms are fundamental in analyzing the efficiency of algorithms. The time complexity of many algorithms is expressed in logarithmic terms, such as O(log n).

4. Statistics

Logarithmic transformations are used to normalize skewed data, making it easier to analyze. Logarithmic regression is used to model relationships where the dependent variable changes exponentially with the independent variable.

Practice Problems

To reinforce your understanding of extraneous solutions in logarithmic equations, here are a few practice problems:

  1. Solve for x: $\log_2(x + 3) + \log_2(x - 1) = 2$
  2. Find the solutions for: $\log_5(2x + 1) - \log_5(x - 2) = 1$
  3. Determine the extraneous solutions, if any: $\log(x^2 - 4) = \log(3x)$

By working through these problems, you will gain confidence in identifying and excluding extraneous solutions.

Conclusion

In conclusion, the concept of extraneous solutions is a vital aspect of solving logarithmic equations. These solutions, though algebraically derived, do not satisfy the original equation due to the domain restrictions of logarithmic functions. The step-by-step solution of the example equation $\log _3(18 x^3) - \log _3(2 x) = \log _3 144$ demonstrated how to identify x = -4 as an extraneous solution. By understanding logarithmic functions, avoiding common mistakes, and recognizing real-world applications, one can master logarithmic equations and the nuances of extraneous solutions. Remember, the key is to always check your solutions in the original equation to ensure their validity.