Solving Logarithmic Equations Step-by-Step Guide For Log(x) - Log(1) = 3
In the realm of mathematics, logarithmic equations often present a unique challenge. However, with a firm grasp of logarithmic properties and algebraic manipulation, these equations can be conquered with confidence. In this comprehensive guide, we will embark on a journey to solve the logarithmic equation log(x) - log(1) = 3, providing a detailed, step-by-step approach that will empower you to tackle similar problems with ease. Our focus will be on transforming this equation into a solvable form by leveraging fundamental logarithmic identities and simplification techniques. By the end of this guide, you will not only possess the solution to the equation but also a deeper understanding of the underlying principles that govern logarithmic functions. We'll explore the properties of logarithms, such as the quotient rule and the logarithm of 1, and demonstrate how these properties are instrumental in simplifying and solving the equation. Moreover, we will emphasize the importance of understanding the domain of logarithmic functions to ensure the validity of the solution. Whether you're a student grappling with logarithmic equations for the first time or a seasoned mathematician seeking a refresher, this guide offers valuable insights and practical strategies for success.
Before diving into the solution, let's establish a solid understanding of logarithmic equations. A logarithmic equation is an equation that involves a logarithm of an expression containing a variable. The logarithm, denoted as log, is the inverse operation of exponentiation. In simpler terms, if we have an equation b^y = x, its logarithmic form is log_b(x) = y, where 'b' is the base, 'x' is the argument, and 'y' is the exponent. When no base is explicitly written, such as in our equation log(x) - log(1) = 3, it is understood to be a common logarithm, which has a base of 10. The key to solving logarithmic equations lies in the astute application of logarithmic properties. These properties allow us to manipulate and simplify the equations, ultimately leading to the isolation of the variable. One of the most crucial properties we'll utilize in this solution is the quotient rule of logarithms, which states that log_b(x) - log_b(y) = log_b(x/y). Additionally, we'll leverage the property that the logarithm of 1 to any base is always 0, i.e., log_b(1) = 0. By understanding these fundamental concepts and properties, we can effectively unravel the complexities of logarithmic equations and arrive at accurate solutions. The domain of a logarithmic function is also crucial to consider. The argument of a logarithm must always be a positive number. This constraint will play a vital role in verifying the validity of our solution later in the process.
Now, let's embark on the journey of solving the equation log(x) - log(1) = 3 step by step. This meticulous approach will not only guide you to the solution but also illustrate the logical progression required for tackling similar logarithmic challenges. By breaking down the problem into manageable steps, we can effectively utilize logarithmic properties and algebraic techniques to simplify the equation and isolate the variable 'x'. Each step will be accompanied by a clear explanation, ensuring that you understand the reasoning behind the transformations. This process will not only provide the answer but also enhance your ability to apply logarithmic principles in various mathematical contexts.
Step 1: Simplify log(1)
The first step in solving the equation log(x) - log(1) = 3 involves simplifying the term log(1). As we discussed earlier, the logarithm of 1 to any base is always 0. This is a fundamental property of logarithms that can significantly simplify equations. In our case, since we are dealing with a common logarithm (base 10), log(1) is indeed equal to 0. This simplification is crucial because it eliminates a term from the equation, making it easier to work with. By substituting log(1) with 0, we transform the equation into a more manageable form that allows us to proceed with further steps in the solution process. This seemingly simple step is a cornerstone of logarithmic problem-solving, highlighting the importance of recognizing and applying key logarithmic properties.
log(x) - log(1) = 3
log(x) - 0 = 3
log(x) = 3
Step 2: Convert to Exponential Form
Having simplified the equation to log(x) = 3, the next crucial step is to convert it from logarithmic form to exponential form. This transformation is the key to isolating the variable 'x' and ultimately solving the equation. Recall that a logarithmic equation log_b(x) = y can be rewritten in exponential form as b^y = x. In our case, the base 'b' is understood to be 10 (since it's a common logarithm), 'x' is the variable we're solving for, and 'y' is 3. Applying this transformation, we rewrite log(x) = 3 as 10^3 = x. This step effectively eliminates the logarithm, allowing us to express 'x' directly in terms of a power of 10. The exponential form provides a clear path to the solution, as we can now directly calculate the value of 'x' by evaluating 10^3. Understanding the relationship between logarithms and exponentials is fundamental to solving logarithmic equations, and this step exemplifies the power of that relationship.
log(x) = 3
10^3 = x
Step 3: Calculate x
With the equation now in the form 10^3 = x, the final step is to calculate the value of x. This involves simply evaluating the exponential expression 10^3, which means 10 multiplied by itself three times. The calculation is straightforward: 10 * 10 * 10 = 1000. Therefore, the solution to the equation is x = 1000. This step brings us to the culmination of the solution process, where we arrive at the numerical value of the variable. It underscores the importance of basic arithmetic skills in the context of solving more complex mathematical problems. The value of x obtained here is the solution to the original logarithmic equation, but it's crucial to verify this solution to ensure its validity within the domain of the logarithmic function.
x = 10^3
x = 1000
Before declaring our solution as final, it's imperative to verify it. This step ensures that the value we obtained for 'x' is indeed a valid solution and not an extraneous one. Extraneous solutions can arise in logarithmic equations due to the domain restrictions of logarithmic functions. The argument of a logarithm must always be a positive number. Therefore, we need to ensure that substituting x = 1000 back into the original equation, log(x) - log(1) = 3, does not result in taking the logarithm of a non-positive number. In our case, substituting x = 1000 gives us log(1000) - log(1) = 3. Since both 1000 and 1 are positive, we are not violating any domain restrictions. Now, let's evaluate the left-hand side of the equation. We know that log(1000) is 3 (since 10^3 = 1000) and log(1) is 0. So, the equation becomes 3 - 0 = 3, which is true. This verification process confirms that x = 1000 is indeed a valid solution to the original logarithmic equation. The importance of verification cannot be overstated, as it safeguards against accepting extraneous solutions and ensures the accuracy of our mathematical work.
log(1000) - log(1) = 3
3 - 0 = 3
3 = 3
After meticulously solving the equation and rigorously verifying our solution, we arrive at the final answer: x = 1000. This value satisfies the original equation log(x) - log(1) = 3 and adheres to the domain restrictions of logarithmic functions. The journey to this solution has been a testament to the power of logarithmic properties and algebraic manipulation. We began by simplifying the equation using the property that log(1) equals 0. Then, we converted the logarithmic equation to its equivalent exponential form, which allowed us to isolate the variable 'x'. Finally, we calculated the value of x and verified that it is a valid solution. This comprehensive approach not only provides the answer but also reinforces the understanding of logarithmic principles and problem-solving techniques. The final answer, x = 1000, represents the culmination of our efforts and a testament to the efficacy of our step-by-step approach.
In conclusion, solving the logarithmic equation log(x) - log(1) = 3 has been a rewarding exercise in applying mathematical principles. We've navigated through the steps of simplification, conversion to exponential form, and verification, ultimately arriving at the solution x = 1000. This process has not only demonstrated the mechanics of solving logarithmic equations but also highlighted the importance of understanding logarithmic properties and domain restrictions. The ability to solve logarithmic equations is a valuable skill in various fields, including mathematics, science, and engineering. The systematic approach we've employed here can be applied to a wide range of logarithmic problems, empowering you to tackle them with confidence and precision. Remember, the key to success lies in a solid understanding of the underlying concepts and a methodical approach to problem-solving. As you continue your mathematical journey, the principles learned here will serve as a strong foundation for tackling more complex challenges. Keep practicing, keep exploring, and continue to unlock the fascinating world of mathematics.
