Expressing Log₁₀(14/3) In Terms Of X, Y, And Z

Hey guys! Let's dive into this logarithmic problem where we're given log₁₀(7) = x, log₁₀(2) = y, and log₁₀(3) = z. Our mission, should we choose to accept it, is to express log₁₀(14/3) in terms of these variables. It sounds like a puzzle, right? But trust me, it's gonna be fun! We'll break it down step by step, making sure everyone's on board.

Understanding the Basics of Logarithms

Before we jump into the solution, let's refresh our understanding of logarithms. Logarithms, at their core, are the inverse operation to exponentiation. Think of it like this: if 10² = 100, then log₁₀(100) = 2. The logarithm tells us what power we need to raise the base (in this case, 10) to get a certain number. Logarithms are super handy for dealing with very large or very small numbers, and they pop up all over the place in science and engineering. There are a few key properties we'll be using, and they're like the secret sauce to solving log problems. One of the most important properties is the product rule, which states that logₐ(mn) = logₐ(m) + logₐ(n). In simpler terms, the logarithm of a product is the sum of the logarithms. Then there's the quotient rule, which is logₐ(m/n) = logₐ(m) - logₐ(n). This means the logarithm of a quotient is the difference of the logarithms. And let's not forget the power rule: logₐ(mᵖ) = p * logₐ(m). This one says the logarithm of a number raised to a power is the power times the logarithm of the number. These rules are like our tools for this problem, and we'll use them to manipulate the expression and get it into the form we want. Remember, the base of the logarithm is 10 in our case, which is super common and often called the common logarithm. So, with these tools in our toolbox, let's tackle the problem at hand and see how we can express log₁₀(14/3) using x, y, and z. It's like we're decoding a secret message, and the logarithmic properties are our decoder ring!

Breaking Down log₁₀(14/3)

Okay, so our main goal here is to express log₁₀(14/3) in terms of x, y, and z. Remember, we know that log₁₀(7) = x, log₁₀(2) = y, and log₁₀(3) = z. The first thing we should do is look at the expression log₁₀(14/3) and think about how we can simplify it using the logarithmic properties we just discussed. The quotient rule looks particularly useful here, right? It says that logₐ(m/n) = logₐ(m) - logₐ(n). So, let's apply that to our expression. We can rewrite log₁₀(14/3) as log₁₀(14) - log₁₀(3). Awesome! We've already made some progress. Now, we have log₁₀(14) - log₁₀(3), and we know what log₁₀(3) is – it's just z. So, we can substitute that in. But what about log₁₀(14)? We need to break that down further. Think about the factors of 14. It's 2 times 7, right? That's perfect because we know the logarithms of both 2 and 7. This is where the product rule comes in handy. The product rule tells us that logₐ(mn) = logₐ(m) + logₐ(n). So, we can rewrite log₁₀(14) as log₁₀(2 * 7). Now, applying the product rule, we get log₁₀(2) + log₁₀(7). This is fantastic because we know log₁₀(2) is y and log₁₀(7) is x. So, log₁₀(14) becomes x + y. We're almost there! Now we can substitute this back into our original expression. We had log₁₀(14/3) = log₁₀(14) - log₁₀(3). We've found that log₁₀(14) = x + y and log₁₀(3) = z. So, let's put it all together. This is like solving a puzzle where all the pieces are finally fitting into place. You can almost feel the solution within your grasp, and this is the best feeling when dealing with math problems, isn't it?

Putting It All Together

Alright, let's bring it all home and express log₁₀(14/3) in terms of x, y, and z. We've done the hard work of breaking down the problem, and now it's time to assemble the final answer. Remember how we started by using the quotient rule to rewrite log₁₀(14/3) as log₁₀(14) - log₁₀(3)? That was a key step. Then, we realized that we needed to simplify log₁₀(14). We factored 14 into 2 * 7, and that allowed us to use the product rule. We transformed log₁₀(14) into log₁₀(2 * 7), which then became log₁₀(2) + log₁₀(7). And we know the values of these! log₁₀(2) is y, and log₁₀(7) is x. So, log₁₀(14) is simply x + y. Now, let's not forget about log₁₀(3). We were given that log₁₀(3) = z. So, we have all the pieces we need. Going back to our original expression, log₁₀(14/3) = log₁₀(14) - log₁₀(3), we can now substitute in what we've found. log₁₀(14) is x + y, and log₁₀(3) is z. So, the expression becomes (x + y) - z. And there we have it! We've successfully expressed log₁₀(14/3) in terms of x, y, and z. It's like we've cracked the code, and the answer is so satisfying. The final expression is x + y - z. This means that the correct answer from the options you provided is C) x + y - z. Awesome job, guys! We took a seemingly complex problem and broke it down into manageable steps, using the properties of logarithms to guide us. Remember, the key to solving these problems is to take it slow, break it down, and use the tools (the logarithmic properties) at your disposal. And always double-check your work to make sure everything adds up. Now you're ready to tackle more logarithmic challenges!

Final Answer

So, the final answer to the question “What is log₁₀(14/3) expressed in terms of x, y, and z, given log₁₀(7) = x, log₁₀(2) = y, and log₁₀(3) = z?” is:

C) x + y - z