Solving Logarithmic Equations Step-by-Step: Find X In Log₈(16) + 2log₈(x) = 2

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Introduction

In the realm of mathematics, logarithmic equations present a unique challenge that demands a methodical approach. Logarithmic equations are equations in which a logarithm appears. These equations often require a deep understanding of logarithmic properties and algebraic manipulation to arrive at the correct solution. This article delves into the process of solving a specific logarithmic equation, providing a detailed step-by-step solution and explaining the underlying principles involved. We will break down each step to ensure a clear understanding of how to tackle such problems effectively. Understanding these equations is crucial not only for academic success in mathematics but also for various applications in science, engineering, and finance. Logarithmic functions are used extensively in areas like calculating acidity (pH levels), measuring sound intensity (decibels), and modeling exponential growth or decay. Therefore, mastering the techniques to solve logarithmic equations is a valuable skill.

This article aims to provide a comprehensive guide to solving the equation log₈(16) + 2log₈(x) = 2, and similar logarithmic problems. We will start by reviewing the essential properties of logarithms, such as the power rule, the product rule, and the change of base formula. Then, we will apply these properties to simplify the given equation and isolate the variable. The process will involve converting logarithmic expressions into exponential forms and using algebraic techniques to find the value of x. This detailed explanation is designed to equip you with the knowledge and skills to confidently solve logarithmic equations, whether you are a student preparing for an exam or a professional applying mathematical concepts in your field. The article will emphasize the importance of checking solutions in the original equation to avoid extraneous roots, which can sometimes arise when dealing with logarithmic functions. By the end of this article, you should have a strong grasp of how to approach and solve logarithmic equations systematically.

Problem Statement

The problem at hand is to find the solution to the logarithmic equation: $\log _8 16+2 \log _8 x=2$. This equation combines a constant logarithmic term with a variable logarithmic term, set equal to a constant. The goal is to isolate x and determine its value that satisfies the equation. Before we proceed with the solution, let’s identify the key components and consider the strategies we can use. We have a base-8 logarithm, which means we are dealing with powers of 8. The equation involves both numerical constants and the variable x within logarithmic functions, which requires us to apply logarithmic properties to simplify and solve for x. This particular equation is a typical example of a problem that combines logarithmic simplification and algebraic manipulation, making it a perfect candidate for demonstrating the application of various logarithmic properties. We need to simplify the equation using properties such as the power rule, which allows us to move coefficients within a logarithm as exponents, and then convert the logarithmic form to exponential form to solve for x. The systematic approach we will follow ensures that we address each aspect of the equation methodically, leading to an accurate solution.

Step-by-Step Solution

Step 1: Simplify the constant term

First, let's simplify the constant term log⁡816\log _8 16. We need to find the power to which we must raise 8 to get 16. Since 16=84/316 = 8^{4/3} or 16=24=(23)4/3=84/316 = 2^4 = (2^3)^{4/3} = 8^{4/3}, we have log⁡816=43\log _8 16 = \frac{4}{3}. Alternatively, we can express 16 as 242^4 and 8 as 232^3, so log⁡816=log⁡2324\log _8 16 = \log _{2^3} 2^4. Using the change of base formula, this is equal to log⁡224log⁡223=43\frac{\log_2 2^4}{\log_2 2^3} = \frac{4}{3}. Thus, the equation becomes: $\frac{4}{3}+2 \log _8 x=2$This initial simplification helps in reducing the complexity of the equation, making the subsequent steps easier to handle. By evaluating the constant logarithmic term, we create a clearer path toward isolating the variable term involving x. This is a crucial step in solving any logarithmic equation, as it sets the stage for further simplification and algebraic manipulation. The ability to recognize and simplify such terms efficiently is a fundamental skill in logarithmic problem-solving.

Step 2: Isolate the logarithmic term

Next, we isolate the term with the variable. Subtract 43\frac{4}{3} from both sides of the equation:$2 \log _8 x=2-\frac4}{3}$$2 \log _8 x=\frac{6}{3}-\frac{4}{3}$$2 \log _8 x=\frac{2}{3}$Now, divide both sides by 2$\log _8 x=\frac{1{3}$This step is essential as it brings us closer to solving for x by separating the variable-containing term from constants. Isolating the logarithmic term allows us to directly apply the definition of logarithms in the next step. The process of isolating the term involves basic algebraic operations, which are crucial in simplifying the equation and paving the way for converting it into exponential form. The goal here is to make the logarithmic expression as simple as possible before converting it to an exponential equation.

Step 3: Convert to exponential form

Now, convert the logarithmic equation to exponential form. The equation log⁡8x=13\log _8 x = \frac{1}{3} means that 813=x8^{\frac{1}{3}}=x. Converting logarithmic equations to exponential form is a core technique in solving these types of problems. It allows us to rewrite the equation in a way that directly involves the variable x, making it easier to solve. The ability to fluidly convert between logarithmic and exponential forms is a critical skill in mathematics, particularly when dealing with logarithmic functions and equations. This step sets the stage for the final calculation, where we evaluate the exponential expression to find the value of x.

Step 4: Solve for x

Evaluate 8138^{\frac{1}{3}}. This is the cube root of 8, which is 2, since 23=82^3 = 8. Therefore, x=2x=2. The cube root of 8 is indeed 2, as 2 multiplied by itself three times equals 8. This calculation directly solves for x once the equation has been converted to exponential form. Evaluating such expressions is often straightforward, involving basic arithmetic and an understanding of roots and powers. This final step in the solution process provides the numerical answer to the problem, representing the value of x that satisfies the original logarithmic equation.

Final Answer

Thus, the solution to the equation is $x=2$. Therefore, the correct answer is D. x=2. This final answer is the culmination of the step-by-step process we followed, demonstrating the systematic approach to solving logarithmic equations. The solution, x=2, is the value that, when substituted back into the original equation, makes the equation true. The verification step, which we will discuss next, is crucial to ensure the accuracy of our solution.

Verification

To verify our solution, substitute x=2x=2 back into the original equation:$\log _8 16+2 \log _8 2=2$We know log⁡816=43\log _8 16 = \frac{4}{3} and log⁡82=13\log _8 2 = \frac{1}{3}, so:$\frac{4}{3}+2\left(\frac{1}{3}\right)=2$$\frac{4}{3}+\frac{2}{3}=2$$\frac{6}{3}=2$$2=2$Since the equation holds true, our solution x=2x=2 is correct. Verification is an essential step in solving mathematical equations, particularly those involving logarithms and radicals, as it helps in identifying and eliminating extraneous solutions. By substituting the solution back into the original equation, we confirm that it satisfies the equation's conditions, thereby ensuring the accuracy and validity of our result. This practice reinforces the understanding of the equation and the solution process, contributing to a deeper comprehension of the mathematical concepts involved.

Common Mistakes to Avoid

When solving logarithmic equations, there are several common mistakes that students often make. One frequent error is incorrectly applying logarithmic properties, such as the power rule or the product rule. For instance, students might mistakenly try to simplify expressions like log₈(16) + 2log₈(x) by directly combining the terms without properly accounting for the coefficient 2. Another common mistake is failing to isolate the logarithmic term before converting to exponential form. Isolating the logarithmic term is crucial because it simplifies the conversion process and reduces the likelihood of errors. Forgetting this step can lead to incorrect application of the definition of logarithms, resulting in an incorrect solution.

Another significant error is neglecting to check for extraneous solutions. Logarithmic functions have domain restrictions, meaning that the argument of a logarithm must be positive. When solving logarithmic equations, we may obtain solutions that do not satisfy these domain restrictions, leading to extraneous solutions. It is essential to substitute the obtained solutions back into the original equation to verify that they are valid. Failing to do so can result in accepting an incorrect answer. Additionally, students sometimes make mistakes in the algebraic manipulation required to solve the equation, such as incorrect arithmetic operations or improper simplification of fractions. To avoid these mistakes, it is essential to practice a systematic approach, carefully follow each step, and double-check calculations. Understanding the properties of logarithms and applying them correctly is also crucial for successful problem-solving.

Conclusion

In summary, solving the logarithmic equation log⁡816+2log⁡8x=2\log _8 16+2 \log _8 x=2 involves a series of steps that require a solid understanding of logarithmic properties and algebraic manipulation. First, we simplified the constant logarithmic term, then we isolated the term containing the variable, converted the equation to exponential form, and finally solved for x. The solution we found was x=2x=2, which we verified by substituting it back into the original equation. This step-by-step process not only provides the solution but also reinforces the principles of solving logarithmic equations. By breaking down the problem into manageable steps, we can approach complex logarithmic equations with confidence. Understanding the properties of logarithms, such as the power rule, the product rule, and the quotient rule, is essential for simplifying and solving these equations effectively.

Furthermore, the ability to convert between logarithmic and exponential forms is a fundamental skill that is crucial for solving logarithmic problems. This conversion allows us to rewrite the equation in a way that makes it easier to isolate and solve for the variable. Avoiding common mistakes, such as incorrectly applying logarithmic properties or neglecting to check for extraneous solutions, is also crucial for achieving accurate results. By practicing these techniques and understanding the underlying concepts, you can confidently solve a wide range of logarithmic equations. The systematic approach outlined in this article can serve as a valuable tool for tackling logarithmic problems in various mathematical contexts.