Solving The Logarithmic Equation Log(x/2) + Log(20/x^2) = Log 8

In this article, we will delve into solving the logarithmic equation presented by Maria: $\log \left(\frac{x}{2}\right)+\log \left(\frac{20}{x^2}\right)=\log 8$. This equation combines logarithmic terms, requiring us to utilize logarithmic properties and algebraic manipulation to find the solution. We will break down each step, ensuring a clear understanding of the process. Logarithmic equations, like this one, often appear in mathematics and various scientific fields, so mastering the techniques to solve them is crucial. The correct solution will be revealed through a detailed, step-by-step approach, making it easy to follow and comprehend the underlying principles. We will explore how to combine logarithms, simplify expressions, and ultimately isolate the variable x. This journey will not only provide the answer but also solidify your understanding of logarithmic functions and their applications.

The first step in tackling this equation is to use the key properties of logarithms to combine the terms on the left-hand side. Specifically, we will use the property that states the sum of logarithms is equal to the logarithm of the product. Mathematically, this property is expressed as $\log_b(m) + \log_b(n) = \log_b(mn)$. Applying this property to Maria's equation, we get:

$\log \left(\frac{x}{2}\right)+\log \left(\frac{20}{x^2}\right) = \log \left(\frac{x}{2} \cdot \frac{20}{x^2}\right)$

Now, we simplify the expression inside the logarithm:

$\log \left(\frac{x}{2} \cdot \frac{20}{x^2}\right) = \log \left(\frac{20x}{2x^2}\right) = \log \left(\frac{10}{x}\right)$

So, our equation now looks like this:

$\log \left(\frac{10}{x}\right) = \log 8$

This simplification is a crucial step. By combining the logarithms, we reduce the complexity of the equation, making it easier to work with. The application of logarithmic properties is a fundamental skill in solving logarithmic equations. Next, we will see how to eliminate the logarithms to solve for x.

With the equation now simplified to $\log \left(\frac{10}{x}\right) = \log 8$, we can proceed to the next step in solving for x. Since the logarithms on both sides of the equation have the same base (base 10, if no base is explicitly written), we can equate the arguments inside the logarithms. This means that if $\log_b(m) = \log_b(n)$, then m = n. Applying this principle to our equation, we have:

$\frac{10}{x} = 8$

Now, we have a simple algebraic equation to solve for x. To isolate x, we can multiply both sides of the equation by x:

$10 = 8x$

Next, we divide both sides by 8:

$x = \frac{10}{8}$

Simplifying the fraction, we get:

$x = \frac{5}{4}$

Thus, we have found a potential solution for x. It is always important to verify the solution in the original equation to ensure it is valid and does not result in taking the logarithm of a negative number or zero. In this case, x = 5/4 is a positive number, so it is a valid solution.

To ensure our solution is correct, we must verify that x = 5/4 satisfies the original equation: $\log \left(\frac{x}{2}\right)+\log \left(\frac{20}{x^2}\right)=\log 8$. Substituting x = 5/4 into the equation, we get:

$\log \left(\frac{5/4}{2}\right)+\log \left(\frac{20}{(5/4)^2}\right)$

First, let's simplify the fractions inside the logarithms:

$\frac{5/4}{2} = \frac{5}{4} \cdot \frac{1}{2} = \frac{5}{8}$

$\left(\frac{5}{4}\right)^2 = \frac{25}{16}$

$\frac{20}{(5/4)^2} = \frac{20}{25/16} = 20 \cdot \frac{16}{25} = \frac{320}{25} = \frac{64}{5}$

Now, substitute these simplified values back into the equation:

$\log \left(\frac{5}{8}\right)+\log \left(\frac{64}{5}\right)$

Again, using the property $\log_b(m) + \log_b(n) = \log_b(mn)$, we combine the logarithms:

$\log \left(\frac{5}{8} \cdot \frac{64}{5}\right) = \log \left(\frac{5 \cdot 64}{8 \cdot 5}\right) = \log \left(\frac{64}{8}\right) = \log 8$

Since the left-hand side simplifies to $\log 8$, which is equal to the right-hand side of the original equation, our solution x = 5/4 is indeed correct. This verification step is essential in solving any equation, especially logarithmic ones, to ensure that the solution is valid and consistent with the original equation.

Therefore, the solution to Maria's equation $\log \left(\frac{x}{2}\right)+\log \left(\frac{20}{x^2}\right)=\log 8$ is x = 5/4. This solution was obtained by first applying the properties of logarithms to combine the logarithmic terms, then simplifying the equation, and finally solving for x. We also verified the solution by substituting it back into the original equation to ensure its validity. The process of solving logarithmic equations involves understanding and applying logarithmic properties, algebraic manipulation, and verification of the solutions.

The answer is (C) $x=\frac{5}{4}$. This problem illustrates the importance of understanding logarithmic properties and algebraic techniques in solving mathematical equations. By mastering these skills, you can tackle a wide range of logarithmic problems and applications in various fields. This comprehensive guide has provided a clear and detailed approach to solving Maria's equation, highlighting the key steps and principles involved in the process. Remember, practice is key to mastering these concepts. Keep practicing, and you will become proficient in solving logarithmic equations and other mathematical challenges. Solving complex equations becomes easier with practice and a solid understanding of the underlying principles. This example serves as a great exercise to reinforce your skills in logarithmic algebra and problem-solving.

What is the solution to the equation $\log \left(\frac{x}{2}\right)+\log \left(\frac{20}{x^2}\right)=\log 8$?