Solving Logarithmic Equations: A Step-by-Step Guide

Introduction to Logarithmic Equations

Do you find logarithmic equations intimidating? Don't worry, you're not alone! Many students grapple with these types of equations, but with a clear understanding of the underlying principles, they can become surprisingly straightforward. In this guide, we'll break down the process of solving a specific logarithmic equation: $\log _2(3x-7)=3$. We'll explore the fundamental concepts, walk through the solution step-by-step, and provide insights to help you tackle similar problems with confidence. Understanding the solution of logarithmic equations is a crucial skill in mathematics, with applications spanning various fields, including physics, engineering, and computer science. Before we dive into the solution, let's recap the basic idea of logarithms. A logarithm answers the question: "To what power must we raise the base to get a certain number?" For instance, $\log_2 8 = 3$ because 2 raised to the power of 3 equals 8. Mastering logarithms opens doors to solving complex problems and gaining a deeper appreciation for the interconnectedness of mathematical concepts. Whether you're a student preparing for an exam or simply curious about expanding your mathematical knowledge, this guide will equip you with the tools and understanding you need to succeed. So, let's embark on this mathematical journey together and unlock the secrets of logarithmic equations!

Understanding the Logarithmic Equation: $\log _2(3x-7)=3$

Before we jump into solving, let's dissect the equation $\log _2(3x-7)=3$ to ensure we grasp what it's telling us. This equation involves a logarithm with base 2. Remember, the base is the small number written below the "log." The expression inside the parentheses, (3x-7), is the argument of the logarithm. The equation states that the logarithm base 2 of (3x-7) is equal to 3. In simpler terms, it's asking: "To what power must we raise 2 to get (3x-7)?" The answer, according to the equation, is 3. This understanding is crucial because it allows us to translate the logarithmic equation into its equivalent exponential form, which is the key to solving it. Recognizing the different parts of the equation – the base, the argument, and the result – is the first step towards unraveling its solution. We can visualize this relationship as 2 raised to the power of 3 equals (3x-7). This transformation from logarithmic to exponential form is a fundamental technique in solving logarithmic equations. By mastering this conversion, you'll be able to approach a wide range of logarithmic problems with greater ease and confidence. The ability to interpret and manipulate logarithmic expressions is a valuable skill in mathematics and beyond, enabling you to tackle challenges in various fields that rely on exponential and logarithmic relationships. So, let's continue our exploration by converting this logarithmic equation into its exponential counterpart.

Converting to Exponential Form

The core strategy for solving logarithmic equations is to convert them into their equivalent exponential form. This conversion allows us to eliminate the logarithm and work with a more familiar algebraic structure. Our equation, $\log _2(3x-7)=3$, can be rewritten in exponential form using the definition of a logarithm. Recall that $\log_b a = c$ is equivalent to $b^c = a$. Applying this to our equation, we identify the base as 2, the exponent as 3, and the result as (3x-7). Therefore, the exponential form of the equation is $2^3 = 3x-7$. This transformation is a pivotal step because it turns a logarithmic equation into a simple algebraic equation that we can solve using standard techniques. By understanding this conversion process, you can effectively tackle a wide array of logarithmic problems. The ability to switch between logarithmic and exponential forms is a cornerstone of logarithmic problem-solving, providing a powerful tool for simplifying equations and finding solutions. Once we have the equation in exponential form, the next step is to simplify and isolate the variable. This involves performing basic arithmetic operations and algebraic manipulations to get the variable, x, by itself on one side of the equation. So, let's proceed with simplifying the exponential form and bringing us closer to the solution.

Simplifying the Exponential Equation

Now that we've converted the logarithmic equation to its exponential form, $2^3 = 3x-7$, the next step is to simplify it. We begin by evaluating $2^3$, which is 2 multiplied by itself three times: $2 * 2 * 2 = 8$. So, our equation becomes $8 = 3x - 7$. This simplified equation is much easier to work with. Our goal now is to isolate the term with 'x' on one side of the equation. To do this, we can add 7 to both sides of the equation. This maintains the equality and moves the constant term to the left side. Adding 7 to both sides gives us $8 + 7 = 3x - 7 + 7$, which simplifies to $15 = 3x$. We're getting closer to solving for 'x'! The key here is to perform the same operations on both sides of the equation to preserve the balance. This principle is fundamental to solving algebraic equations and ensures that the solution we find is valid. By simplifying the equation step-by-step, we're making it easier to see the next steps required to isolate the variable. The next logical step is to get 'x' completely by itself, which we can achieve by dividing both sides of the equation by the coefficient of 'x'. So, let's move on to the final step of isolating 'x' and finding the solution.

Isolating 'x' and Finding the Solution

We've simplified the equation to $15 = 3x$. To isolate 'x', we need to get rid of the coefficient 3 that's multiplying it. We can do this by dividing both sides of the equation by 3. This gives us $\frac{15}{3} = \frac{3x}{3}$, which simplifies to $5 = x$. Therefore, the solution to the equation is $x = 5$. But we're not quite done yet! It's crucial to check our solution in the original logarithmic equation to make sure it's valid. This is because logarithmic functions have domain restrictions; the argument of a logarithm must be positive. Plugging $x = 5$ back into the original equation, $\log _2(3x-7)=3$, we get $\log _2(3(5)-7)=3$. Simplifying the argument, we have $\log _2(15-7)=3$, which becomes $\log _2(8)=3$. Since $2^3 = 8$, this confirms that our solution is correct. Checking the solution is an essential step in solving logarithmic equations, as it ensures that we haven't introduced any extraneous solutions. In this case, $x = 5$ is a valid solution, meaning it satisfies the original equation and lies within the domain of the logarithmic function. Now that we've found and verified the solution, we can confidently say that the answer to the equation $\log _2(3x-7)=3$ is $x = 5$.

Conclusion

In this guide, we've walked through the process of solving the logarithmic equation $\log _2(3x-7)=3$. We started by understanding the equation and its components, then converted it to exponential form. After simplifying the exponential equation, we isolated 'x' and found the solution to be $x = 5$. Finally, we checked our solution in the original equation to ensure its validity. Solving logarithmic equations requires a clear understanding of logarithms and their relationship to exponential functions. By mastering the techniques outlined in this guide, you'll be well-equipped to tackle a variety of logarithmic problems. Remember, the key is to convert the logarithmic equation to its exponential form, simplify, and then isolate the variable. And always remember to check your solution! Practice is essential for building confidence and proficiency in solving logarithmic equations. Work through various examples and challenge yourself with increasingly complex problems. With dedication and a solid understanding of the underlying concepts, you can conquer logarithmic equations and expand your mathematical toolkit. For further exploration and practice, you might find the resources available at Khan Academy's Logarithm Section helpful.