Solving Logarithmic Equations A Step-by-Step Guide

In this article, we will delve into the process of solving a logarithmic equation. Logarithmic equations are a fundamental part of mathematics, appearing in various fields such as physics, engineering, and computer science. The specific equation we will tackle is: log(x+10) - log(x) = 2. This equation involves logarithms with the same base, making it a classic example that can be solved using the properties of logarithms. Our approach will be systematic, ensuring that each step is clearly explained and justified. We'll begin by understanding the basic properties of logarithms, then apply these properties to simplify the equation, and finally, solve for the unknown variable, x. Furthermore, we will verify our solution to ensure it is valid, as logarithmic equations can sometimes yield extraneous solutions. By the end of this article, you will have a comprehensive understanding of how to solve this type of logarithmic equation and similar problems.

Before we dive into solving the equation, it is essential to understand the properties of logarithms that we will be using. Logarithms are the inverse operations of exponentiation. The logarithm of a number to a given base is the exponent to which the base must be raised to produce that number. Mathematically, if b^y = x, then log_b(x) = y, where b is the base, x is the argument, and y is the logarithm. The equation we are dealing with uses logarithms without an explicitly stated base. In such cases, it is conventionally understood that the base is 10, known as the common logarithm. Therefore, log(x) is equivalent to log_10(x). One crucial property of logarithms that we will use in solving the equation is the quotient rule. The quotient rule states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. Mathematically, this is expressed as: log_b(m/n) = log_b(m) - log_b(n). This property is the key to simplifying the left side of our equation. Another important concept is the relationship between logarithms and exponentiation. To remove a logarithm from an equation, we can exponentiate both sides of the equation using the base of the logarithm. For instance, if log_b(x) = y, then b^log_b(x) = b^y, which simplifies to x = b^y. This step is essential for isolating the variable x and finding its value. With these properties in mind, we are well-equipped to solve the given logarithmic equation.

Now, let's apply the properties of logarithms to solve the equation log(x+10) - log(x) = 2. Our first step is to use the quotient rule of logarithms to combine the two logarithmic terms on the left side of the equation. According to the quotient rule, log(x+10) - log(x) can be rewritten as log((x+10)/x). Thus, our equation becomes log((x+10)/x) = 2. Next, we need to eliminate the logarithm. Since the base of the logarithm is 10, we can exponentiate both sides of the equation with base 10. This means we raise 10 to the power of both sides of the equation: 10^(log((x+10)/x)) = 10^2. The left side simplifies to (x+10)/x, and the right side simplifies to 100. So, our equation now is (x+10)/x = 100. To solve for x, we need to isolate it. First, we multiply both sides of the equation by x to get rid of the fraction: x * (x+10)/x = 100 * x, which simplifies to x+10 = 100x. Next, we subtract x from both sides of the equation to gather the x terms on one side: x+10 - x = 100x - x, which simplifies to 10 = 99x. Finally, we divide both sides by 99 to solve for x: 10/99 = 99x/99, which gives us x = 10/99. Therefore, the solution to the equation is x = 10/99. However, it is crucial to verify this solution to ensure it is valid.

After finding a potential solution to a logarithmic equation, it is essential to verify it. This is because logarithms are only defined for positive arguments. If our solution leads to a negative or zero argument inside any logarithm in the original equation, it is considered an extraneous solution and must be discarded. Our solution for the equation log(x+10) - log(x) = 2 is x = 10/99. To verify this, we need to substitute this value back into the original equation and check if it holds true. First, let's substitute x = 10/99 into the argument of the first logarithm, x+10. This gives us (10/99) + 10. To add these, we need a common denominator, so we rewrite 10 as 990/99. Thus, (10/99) + (990/99) = 1000/99. Since 1000/99 is a positive number, the first logarithm, log(x+10), is defined. Next, let's consider the argument of the second logarithm, x. Since x = 10/99, which is a positive number, the second logarithm, log(x), is also defined. Now, we need to check if the equation holds true with x = 10/99. Substituting this value into the original equation, we get log((10/99) + 10) - log(10/99) = 2. We already found that (10/99) + 10 = 1000/99, so the equation becomes log(1000/99) - log(10/99) = 2. Using the quotient rule of logarithms, we can rewrite this as log((1000/99) / (10/99)) = 2. Simplifying the fraction inside the logarithm, we get log(1000/10) = 2, which further simplifies to log(100) = 2. Since log_10(100) = 2, the equation holds true. Therefore, our solution x = 10/99 is valid.

In this article, we have successfully solved the logarithmic equation log(x+10) - log(x) = 2. We began by understanding the fundamental properties of logarithms, particularly the quotient rule, which allows us to combine logarithmic terms, and the relationship between logarithms and exponentiation, which enables us to eliminate logarithms from the equation. We then systematically applied these properties to simplify the equation, step-by-step, eventually arriving at the solution x = 10/99. A crucial part of the process was the verification step, where we substituted our solution back into the original equation to ensure it was valid and not an extraneous solution. We found that x = 10/99 indeed satisfies the equation, confirming its validity. This process highlights the importance of not only solving equations but also verifying the solutions, especially in the context of logarithmic and other transcendental equations. By mastering these techniques, you can confidently tackle a wide range of logarithmic equations. The skills and understanding gained here are valuable in various mathematical and scientific contexts, making this a fundamental topic in mathematics education. Remember to always consider the domain of logarithmic functions when solving equations, and to verify your solutions to ensure accuracy.