Evaluate: Log₁₀[1/3 + 1/4] + 2Log₁₀[2 + Log(3/7)]

Hey there, math enthusiasts! Ever stumbled upon a mathematical expression that looks like it belongs in a cryptic puzzle? Well, today, we're diving headfirst into one such intriguing problem. We're going to break down and evaluate the expression: log₁₀[1/3 + 1/4] + 2log₁₀[2 + log(3/7)]. Sounds intimidating? Don't worry, we'll take it step by step, making sure everyone can follow along. So, grab your thinking caps, and let's get started!

Unraveling the First Logarithmic Term: log₁₀[1/3 + 1/4]

Let's kick things off by tackling the first part of our expression: log₁₀[1/3 + 1/4]. Now, before we get lost in the logarithmic jungle, let's simplify what's inside the brackets. We've got a simple addition of two fractions, 1/3 and 1/4. Remember how to add fractions? We need a common denominator, guys! The least common multiple of 3 and 4 is 12. So, we'll convert both fractions to have a denominator of 12. 1/3 becomes 4/12 (multiply both numerator and denominator by 4), and 1/4 becomes 3/12 (multiply both numerator and denominator by 3). Now, we can easily add them: 4/12 + 3/12 = 7/12. Great! We've simplified the expression inside the brackets to 7/12. So, our first term now looks like log₁₀[7/12]. But what does this actually mean? Remember, logarithms are just the inverse of exponentiation. In simpler terms, log₁₀[7/12] asks the question: "To what power must we raise 10 to get 7/12?" Now, we can't just conjure up the answer to that in our heads (unless you're a math wizard, of course!), so we'll need to use a calculator or some logarithmic properties to figure this out. Before we reach for our calculators, let's think about some properties that might help us. One useful property of logarithms is the quotient rule, which states that logₐ(b/c) = logₐ(b) - logₐ(c). This means we can rewrite log₁₀[7/12] as log₁₀(7) - log₁₀(12). This might seem like we're making things more complicated, but trust me, it's a step in the right direction. Now we have two simpler logarithmic terms. We can use a calculator to find the approximate values of log₁₀(7) and log₁₀(12). log₁₀(7) is approximately 0.8451, and log₁₀(12) is approximately 1.0792. So, log₁₀(7/12) ≈ 0.8451 - 1.0792 ≈ -0.2341. We've successfully evaluated the first term! It's approximately -0.2341. Let's keep this value in mind as we move on to the next part of our expression.

Deciphering the Second Logarithmic Term: 2log₁₀[2 + log(3/7)]

Alright, let's shift our focus to the second, and arguably trickier, part of the expression: 2log₁₀[2 + log(3/7)]. This term has a logarithm nested inside another logarithm, which might seem a bit daunting at first. But don't worry, we'll conquer it! The key here is to work from the inside out. So, let's first focus on the innermost logarithm: log(3/7). Notice that there's no base explicitly written for this logarithm. When the base isn't specified, it's generally assumed to be base 10 (that is, log₁₀). So, we're essentially dealing with log₁₀(3/7). Just like before, we can use the quotient rule of logarithms to rewrite this as log₁₀(3) - log₁₀(7). This makes things a bit easier to handle. Now, let's grab our calculators (or logarithmic tables, if you're feeling old-school) and find the approximate values of log₁₀(3) and log₁₀(7). We already know that log₁₀(7) is approximately 0.8451. log₁₀(3) is approximately 0.4771. Therefore, log₁₀(3/7) ≈ 0.4771 - 0.8451 ≈ -0.3680. Okay, we've successfully evaluated the inner logarithm! Now we can substitute this value back into our original term. We now have 2log₁₀[2 + (-0.3680)]. Let's simplify the expression inside the brackets first. 2 + (-0.3680) is simply 2 - 0.3680, which equals 1.6320. So, our term now looks like 2log₁₀(1.6320). We're getting there! Now, let's deal with that coefficient of 2. Remember the power rule of logarithms? It states that logₐ(bᶜ) = c * logₐ(b). We can use this rule in reverse! This means that c * logₐ(b) = logₐ(bᶜ). In our case, we have 2 * log₁₀(1.6320), which can be rewritten as log₁₀(1.6320²). This simplifies our expression quite a bit. Now, we just need to calculate 1.6320². Using a calculator, we find that 1.6320² ≈ 2.6634. So, our term now becomes log₁₀(2.6634). One final step! Let's use our calculator to find the approximate value of log₁₀(2.6634). It's approximately 0.4254. Phew! We've successfully evaluated the second term: 2log₁₀[2 + log(3/7)] ≈ 0.4254. That was a bit of a journey, but we made it through!

Combining the Results: The Grand Finale

We've done the hard work! We've evaluated both parts of our expression separately. Now, the final step is to simply add the results together. We found that log₁₀[1/3 + 1/4] ≈ -0.2341, and 2log₁₀[2 + log(3/7)] ≈ 0.4254. So, let's add them up: -0.2341 + 0.4254 ≈ 0.1913. And there you have it! The approximate value of the expression log₁₀[1/3 + 1/4] + 2log₁₀[2 + log(3/7)] is approximately 0.1913. That's quite a feat of mathematical exploration! We tackled fractions, logarithms, and even nested logarithms. You guys are amazing!

Wrapping Up: The Beauty of Mathematical Problem-Solving

So, what did we learn today? We took a complex-looking logarithmic expression and, by breaking it down into smaller, manageable parts, we were able to evaluate it successfully. We brushed up on our fraction skills, revisited the properties of logarithms (quotient rule and power rule), and used calculators to find approximate values. But more importantly, we learned the value of methodical problem-solving. When faced with a challenging problem, don't be intimidated! Break it down, identify the key concepts, and tackle it step by step. Just like we did with this logarithmic expression, you can conquer any mathematical challenge that comes your way. Keep practicing, keep exploring, and keep enjoying the beauty of mathematics! And remember, there's always a solution to be found, even if it takes a little bit of digging. You've got this!