Solving Log₄(-3x + 8) = 1 A Step-by-Step Guide

Introduction: Understanding Logarithmic Equations

In the realm of mathematics, logarithmic equations often pose a challenge, but with a systematic approach, they can be solved effectively. This article aims to provide a comprehensive guide on how to solve the logarithmic equation log₄(-3x + 8) = 1. We will break down the steps involved, ensuring a clear understanding of the underlying principles and techniques. Understanding logarithmic equations is crucial for various fields, including engineering, physics, computer science, and finance. Logarithms are used to model and solve problems involving exponential growth and decay, such as compound interest, population growth, and radioactive decay. Before diving into the specific problem, let's lay a solid foundation by defining what logarithms are and exploring their fundamental properties. A logarithm is the inverse operation to exponentiation. In simpler terms, if we have an equation like b^y = x, the logarithm (base b) of x is y. This is written as log_b(x) = y. Here, b is the base of the logarithm, x is the argument, and y is the exponent. For instance, if we have 2^3 = 8, then the logarithm base 2 of 8 is 3, which is expressed as log₂(8) = 3. Understanding this relationship is fundamental to solving logarithmic equations. Logarithmic equations come in various forms, but many can be simplified using the properties of logarithms. These properties include the product rule, quotient rule, and power rule. The product rule states that log_b(mn) = log_b(m) + log_b(n), meaning the logarithm of a product is the sum of the logarithms. The quotient rule states that log_b(m/n) = log_b(m) - log_b(n), meaning the logarithm of a quotient is the difference of the logarithms. The power rule states that log_b(m^p) = p * log_b(m), meaning the logarithm of a number raised to a power is the product of the power and the logarithm of the number. These properties allow us to manipulate logarithmic expressions, combine or separate terms, and simplify equations, making them easier to solve. Furthermore, it's essential to understand the domain of logarithmic functions. The argument of a logarithm must be positive because you cannot take the logarithm of a negative number or zero. This is a critical consideration when solving logarithmic equations, as we need to ensure that our solutions do not result in the logarithm of a non-positive number. Keeping these basics in mind, we can now approach the given equation with a clear strategy and confidence. The ability to solve logarithmic equations is not just an academic exercise; it's a valuable skill that can be applied in numerous real-world scenarios. From determining the magnitude of an earthquake using the Richter scale to calculating the pH of a solution in chemistry, logarithms play a crucial role in quantitative analysis and problem-solving.

Step 1: Convert the Logarithmic Equation to Exponential Form

To effectively solve the equation log₄(-3x + 8) = 1, the initial and crucial step involves converting the logarithmic equation into its equivalent exponential form. This transformation allows us to eliminate the logarithm and work with a more manageable algebraic expression. The fundamental relationship between logarithms and exponentials is the key to this conversion. Recall that a logarithmic equation of the form log_b(a) = c is equivalent to the exponential equation b^c = a. Here, 'b' is the base of the logarithm, 'a' is the argument, and 'c' is the result of the logarithm. Applying this principle to our equation, log₄(-3x + 8) = 1, we identify the base as 4, the argument as (-3x + 8), and the result as 1. Therefore, the exponential form of the equation is 4¹ = -3x + 8. This conversion is the cornerstone of solving logarithmic equations because it translates the problem into a more familiar algebraic format. By understanding this relationship, you can easily move between logarithmic and exponential forms, which is essential for manipulating and solving various types of logarithmic problems. The simplicity of the exponential form allows us to apply standard algebraic techniques to isolate the variable 'x'. The equation 4¹ = -3x + 8 is a linear equation, which we can solve using basic algebraic operations. The ability to transform logarithmic equations into exponential forms is not just a mathematical trick; it reflects a deeper understanding of the nature of logarithms and their inverse relationship with exponentials. This understanding is vital in many areas of science and engineering, where logarithmic scales are used to represent a wide range of phenomena, from the intensity of sound to the concentration of chemicals. Furthermore, the process of converting logarithmic equations to exponential forms reinforces the importance of mathematical equivalence. It demonstrates how the same mathematical relationship can be expressed in different forms, each offering its own advantages for analysis and problem-solving. The exponential form often simplifies calculations and provides a clearer picture of the underlying relationships between variables. In the context of solving logarithmic equations, this conversion is often the most critical step. Once the equation is in exponential form, the remaining steps usually involve straightforward algebraic manipulations. However, the initial conversion is the key to unlocking the solution. By mastering this technique, students and professionals alike can approach logarithmic equations with confidence and solve them efficiently.

Step 2: Simplify and Isolate the Variable

Following the conversion of the logarithmic equation into its exponential form, the next step involves simplifying the equation and isolating the variable 'x'. This process requires applying basic algebraic principles to rearrange the terms and solve for 'x'. Our equation, now in exponential form, is 4¹ = -3x + 8. The first simplification we can make is to evaluate 4¹, which is simply 4. Thus, the equation becomes 4 = -3x + 8. To isolate the term containing 'x', we need to eliminate the constant term on the right side of the equation. We can do this by subtracting 8 from both sides of the equation. This maintains the equality and moves us closer to isolating 'x'. Subtracting 8 from both sides, we get 4 - 8 = -3x + 8 - 8, which simplifies to -4 = -3x. Now, the variable 'x' is attached to a coefficient, -3. To isolate 'x', we need to divide both sides of the equation by -3. This will give us the value of 'x'. Dividing both sides by -3, we have -4 / -3 = -3x / -3, which simplifies to x = 4/3. Therefore, we have found a potential solution for 'x'. However, it's crucial to remember that we are working with a logarithmic equation, and we need to verify that this solution is valid. The process of simplifying and isolating the variable 'x' demonstrates the power of algebraic manipulation. By applying basic operations such as addition, subtraction, multiplication, and division, we can rearrange equations and solve for unknowns. This skill is fundamental to mathematics and is used extensively in various fields, including science, engineering, and economics. In the context of solving equations, the goal is to isolate the variable of interest on one side of the equation, so that its value can be determined. This often involves a series of steps, each designed to eliminate terms or coefficients that are preventing the variable from being isolated. The process also highlights the importance of maintaining equality throughout the manipulations. Whatever operation is performed on one side of the equation must also be performed on the other side to ensure that the equation remains balanced. This principle is the cornerstone of algebraic problem-solving. The result, x = 4/3, is a candidate solution, but we must now proceed to the next crucial step: verifying the solution in the original logarithmic equation. This step is particularly important when dealing with logarithmic and radical equations, where extraneous solutions can arise.

Step 3: Verify the Solution

The verification step is crucial in solving logarithmic equations to ensure that the solution obtained is valid and does not lead to any undefined terms in the original equation. Logarithmic functions have a specific domain: the argument of the logarithm must be positive. Therefore, any solution that results in a non-positive argument must be discarded. Our original equation is log₄(-3x + 8) = 1, and we found a potential solution of x = 4/3. To verify this solution, we substitute x = 4/3 back into the argument of the logarithm, which is -3x + 8. Substituting x = 4/3, we get -3(4/3) + 8. Simplifying this expression, we have -4 + 8, which equals 4. Since 4 is positive, the argument of the logarithm is positive when x = 4/3. This means that our solution does not violate the domain of the logarithmic function. Now that we have confirmed that the argument is positive, we can substitute x = 4/3 back into the original equation to see if it holds true. Substituting x = 4/3 into log₄(-3x + 8) = 1, we get log₄(-3(4/3) + 8) = 1. We already calculated that -3(4/3) + 8 = 4, so the equation becomes log₄(4) = 1. By definition, log₄(4) is indeed 1, because 4¹ = 4. Therefore, the solution x = 4/3 satisfies the original equation. This verification process is not merely a formality; it is an essential part of solving logarithmic equations. It ensures that the solution we have found is not an extraneous solution, which can arise due to the properties of logarithms. Extraneous solutions are values that satisfy the transformed equation but not the original equation. They often occur when we perform operations that are not reversible, such as squaring both sides of an equation or, in this case, dealing with the domain restrictions of logarithms. The importance of verification cannot be overstated. It is a fundamental principle in mathematics to check your work and ensure that your solutions are valid. In practical applications, using an incorrect solution can lead to significant errors and consequences. Therefore, always take the time to verify your solutions, especially when dealing with functions that have domain restrictions. In summary, the verification step confirms the validity of our solution, x = 4/3, by ensuring that it results in a positive argument for the logarithm and that it satisfies the original equation. This completes the process of solving the logarithmic equation.

Conclusion: The Solution and Key Takeaways

In conclusion, we have successfully solved the logarithmic equation log₄(-3x + 8) = 1. Through a step-by-step approach, we converted the logarithmic equation to its exponential form, simplified the resulting algebraic equation, and isolated the variable 'x'. We found a potential solution of x = 4/3, and crucially, we verified this solution to ensure its validity. The verification step confirmed that x = 4/3 is indeed the correct solution, as it satisfies the original equation and does not violate the domain restrictions of the logarithmic function. This process highlights several key takeaways for solving logarithmic equations. First, understanding the fundamental relationship between logarithms and exponentials is essential. This relationship allows us to convert logarithmic equations into more manageable algebraic forms. The ability to switch between logarithmic and exponential forms is a cornerstone of solving these types of equations. Second, algebraic manipulation skills are critical. Simplifying equations, isolating variables, and maintaining equality are all fundamental algebraic techniques that are used extensively in solving logarithmic equations. A solid understanding of these techniques is necessary for success. Third, verification is paramount. The verification step ensures that the solution obtained is valid and not an extraneous solution. This step is particularly important when dealing with functions that have domain restrictions, such as logarithms. Always take the time to verify your solutions to avoid errors. Furthermore, the process of solving logarithmic equations reinforces the importance of a systematic approach to problem-solving. By breaking down the problem into smaller, manageable steps, we can tackle complex equations with confidence. Each step builds upon the previous one, leading us to the solution in a logical and organized manner. The skills and concepts learned in solving logarithmic equations are applicable to many other areas of mathematics and science. Logarithms are used in various fields, including engineering, physics, computer science, and finance. Understanding how to solve logarithmic equations is a valuable skill that can be applied in numerous real-world scenarios. In summary, solving log₄(-3x + 8) = 1 involved converting to exponential form, simplifying and isolating 'x', and verifying the solution. The correct solution is x = 4/3. This exercise demonstrates the importance of understanding logarithmic properties, applying algebraic techniques, and the necessity of verifying solutions. By mastering these concepts, you can confidently tackle a wide range of logarithmic equations and related problems.