Solving 5 + Log2(6x) = 8 Step-by-Step Guide

Introduction

In this comprehensive guide, we will delve into the process of solving the logarithmic equation 5 + log₂(6x) = 8. This equation falls under the category of logarithmic equations, which are equations where the variable appears within a logarithmic function. Solving logarithmic equations is a fundamental skill in mathematics, particularly in algebra and calculus, and it has applications in various fields such as finance, physics, and computer science. This article aims to provide a step-by-step solution to the given equation, along with a detailed explanation of the underlying concepts and techniques involved. We will explore the properties of logarithms, the process of isolating the logarithmic term, converting logarithmic equations to exponential form, and finally, solving for the variable x. Furthermore, we will discuss the importance of verifying the solution to ensure it satisfies the original equation and does not result in any undefined terms. By the end of this guide, you will have a solid understanding of how to solve logarithmic equations and be able to apply these techniques to similar problems.

Understanding Logarithmic Equations

Before we dive into the solution, it's essential to understand the basics of logarithmic equations. A logarithm is the inverse operation to exponentiation. In simple terms, if we have an equation bˣ = y, then the logarithm of y to the base b is x, written as log_b(y) = x. The base b is a positive real number not equal to 1, x is the exponent, and y is the result of raising b to the power of x. Understanding this relationship is crucial for solving logarithmic equations. Logarithmic equations often involve isolating the logarithmic term and then converting the equation into exponential form to solve for the variable. This process involves using various properties of logarithms, such as the product rule, quotient rule, and power rule, which allow us to simplify and manipulate logarithmic expressions. Additionally, it's important to remember that the argument of a logarithm (the value inside the parentheses) must be positive, as logarithms are not defined for non-positive values. This constraint plays a critical role in verifying the solutions we obtain. In the context of our equation, 5 + log₂(6x) = 8, we will first isolate the logarithmic term log₂(6x) and then convert the equation into exponential form to solve for x. This will involve a careful application of logarithmic properties and algebraic manipulation.

Step-by-Step Solution

Let's solve the logarithmic equation 5 + log₂(6x) = 8 step-by-step.

Step 1: Isolate the Logarithmic Term

The first step in solving this equation is to isolate the logarithmic term, which is log₂(6x). To do this, we need to subtract 5 from both sides of the equation:

5 + log₂(6x) - 5 = 8 - 5

This simplifies to:

log₂(6x) = 3

Now we have the logarithmic term isolated on one side of the equation.

Step 2: Convert to Exponential Form

Next, we need to convert the logarithmic equation into its equivalent exponential form. Recall that log_b(y) = x is equivalent to bˣ = y. In our case, the base b is 2, the exponent x is 3, and the argument y is 6x. So, we can rewrite the equation as:

2³ = 6x

Step 3: Simplify and Solve for x

Now, we can simplify the equation and solve for x. First, calculate 2³:

2³ = 8

So our equation becomes:

8 = 6x

To solve for x, we divide both sides of the equation by 6:

x = 8 / 6

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

x = 4 / 3

Therefore, the solution for x is 4/3.

Step 4: Verify the Solution

It's crucial to verify our solution to ensure it is valid and does not result in any undefined terms in the original equation. The argument of a logarithm must be positive, so we need to make sure that 6x > 0 when x = 4/3. Let's substitute x = 4/3 into the argument:

6 * (4 / 3) = 8

Since 8 is positive, the logarithm is defined for x = 4/3. Now, let's substitute x = 4/3 back into the original equation to see if it holds true:

5 + log₂(6 * (4 / 3)) = 8

5 + log₂(8) = 8

Since log₂(8) = 3 (because 2³ = 8), we have:

5 + 3 = 8

8 = 8

The equation holds true, so our solution x = 4/3 is valid.

Analyzing the Answer Choices

Now that we have found the solution, let's analyze the provided answer choices:

A) x = 4 / 3 B) x = 3 / 4 C) x = 48 D) x = 18

Our solution, x = 4/3, matches answer choice A.

Common Mistakes to Avoid

When solving logarithmic equations, there are several common mistakes that students often make. Understanding these mistakes can help you avoid them and solve equations more accurately. One common mistake is forgetting to isolate the logarithmic term before converting the equation to exponential form. It is crucial to isolate the logarithmic term first to ensure the conversion is done correctly. Another mistake is incorrectly applying the properties of logarithms. For example, students might incorrectly combine logarithmic terms or apply the power rule improperly. It is essential to have a strong understanding of the logarithmic properties and apply them carefully. A crucial step that is often overlooked is verifying the solution. The argument of a logarithm must be positive, so any solution that results in a non-positive argument is extraneous and must be discarded. Failing to verify the solution can lead to incorrect answers. Another common mistake is making algebraic errors when simplifying the equation. This can include errors in arithmetic, such as incorrect addition or multiplication, or errors in algebraic manipulation, such as incorrectly distributing terms or combining like terms. To avoid these mistakes, it is important to work carefully, show all steps, and double-check your work. Practicing a variety of logarithmic equations can also help you build confidence and avoid common errors. In the context of the equation 5 + log₂(6x) = 8, it's crucial to remember to subtract 5 from both sides before converting to exponential form and to verify that 6x is positive for the solution obtained.

Conclusion

In conclusion, solving the logarithmic equation 5 + log₂(6x) = 8 involves several key steps: isolating the logarithmic term, converting the equation to exponential form, solving for the variable, and verifying the solution. By following these steps carefully, we found that the solution to the equation is x = 4/3. It's important to understand the properties of logarithms and the relationship between logarithmic and exponential forms to solve these types of equations effectively. Additionally, verifying the solution is a crucial step to ensure the answer is valid and does not lead to any undefined terms. This process not only confirms the correctness of the solution but also reinforces the understanding of the domain restrictions of logarithmic functions. Common mistakes, such as failing to isolate the logarithmic term or neglecting to verify the solution, can lead to incorrect answers. Therefore, practicing a variety of logarithmic equations and paying attention to detail are essential for mastering this skill. The ability to solve logarithmic equations is a fundamental mathematical skill that has applications in various fields, making it an important topic to understand thoroughly. With practice and a clear understanding of the underlying concepts, solving logarithmic equations can become a straightforward process.