Solving Logarithm Problems: A Step-by-Step Guide

Hey guys, let's dive into a cool math problem that involves exponents and logarithms! We're given some information about the relationship between a, b, and c, and we need to figure out the value of a specific expression. Don't worry, it's not as scary as it looks. We'll break it down step by step to make it super easy to understand. So, grab your pencils, and let's get started!

Understanding the Problem and Given Information

First things first, let's make sure we totally get what the problem is asking. We're given a few key pieces of information:

• We know that a raised to the power of x (aˣ), b raised to the power of y (bʸ), and c raised to the power of z (cᶻ) all equal 64. This is a super important starting point, so let's keep it in mind.
• We also know that 2 times the logarithm of the product of a, b, and c (abc) equals 6 (2 log(abc) = 6). This is where the logarithms come into play, and it's our key to unlocking the solution. Remember, the logarithm is the inverse of exponentiation, so we'll be using this relationship to solve the problem.

Our ultimate goal is to find the value of 1/x + 1/y + 1/z. This looks a little tricky at first, but we'll see how everything comes together as we work through the problem. This is a classic example of a math problem where understanding the properties of exponents and logarithms is key. Let's start with the first piece of information.

From aˣ = bʸ = cᶻ = 64, we can deduce some important relationships. Since each term equals 64, we can express a, b, and c in terms of 64 and their respective exponents. This will be super helpful later on. Knowing this, we can try to relate the exponents and the value 64. Remember that 64 can be expressed as a power of 2: 64 = 2⁶. This is often the trick in these types of problems – recognizing the underlying relationships between numbers and their prime factors. This understanding helps us simplify the problem and make it easier to solve.

Now, let's focus on the second piece of information: 2 log(abc) = 6. This is where we bring in the logarithms. We can simplify this equation by dividing both sides by 2, which gives us log(abc) = 3. This tells us that the logarithm (base 10) of abc is 3. We can then rewrite this in exponential form: abc = 10³. While this might seem like we're not getting anywhere, it will be useful later when we start relating this back to our first set of equations.

Solving for the Unknown Value: 1/x + 1/y + 1/z

Alright, now for the fun part: finding the value of 1/x + 1/y + 1/z. This is where we bring everything together and use our understanding of exponents and logarithms. Let's start by working with the first set of equations (aˣ = bʸ = cᶻ = 64) and try to express a, b, and c in terms of 64. Remember that we can rewrite 64 as 2⁶. This will allow us to simplify the equations further. For example, if aˣ = 64, we can rewrite this as aˣ = 2⁶.

To find a, we can take the x-th root of both sides. This gives us a = 64^(1/x), or a = 2^(6/x). We can do the same thing for b and c. Since bʸ = 64, then b = 64^(1/y) = 2^(6/y). Similarly, since cᶻ = 64, then c = 64^(1/z) = 2^(6/z).

Now we have a, b, and c expressed in terms of 2 and their respective exponents. The next step is to substitute these values into the equation log(abc) = 3. Remember that abc = a * b * c, so we can substitute the values we just found: log(2^(6/x) * 2^(6/y) * 2^(6/z)) = 3. Using the properties of exponents, when multiplying with the same base, you can add the exponents. This means that 2^(6/x) * 2^(6/y) * 2^(6/z) = 2^(6/x + 6/y + 6/z).

So, our equation becomes log(2^(6/x + 6/y + 6/z)) = 3. Now, remember that log(x) = y can be rewritten as 10ʸ = x. Using this fact, our equation transforms into 2^(6/x + 6/y + 6/z) = 10³. However, notice that we used base 10 for the logarithm; it's not super helpful. We used it to make the equation easier to follow. Instead, let's use the property of logarithms that logₐ(b) = c can be written as aᶜ = b. Recall from the previous section that abc = 10³, but our base here is 2. Let's rewrite log(abc) = 3 as log₂(abc) = 6 to match the base from aˣ = bʸ = cᶻ = 64. So, we'll use the property logₐ(bc) = logₐ(b) + logₐ(c).

Since abc = 2^(6/x + 6/y + 6/z), the equation becomes 2^(6/x + 6/y + 6/z) = 64. Now, remember that we can also express 64 as 2⁶. Thus, our equation becomes 2^(6/x + 6/y + 6/z) = 2⁶. Since the bases are the same, we can equate the exponents: 6/x + 6/y + 6/z = 6. Dividing both sides by 6, we get 1/x + 1/y + 1/z = 1. Therefore, the value of 1/x + 1/y + 1/z is 1.

Conclusion

Great job, everyone! We successfully solved the problem and found that the value of 1/x + 1/y + 1/z is 1. We did this by breaking down the problem step-by-step, using the properties of exponents and logarithms. Understanding these properties is crucial for tackling similar problems. Remember, practice makes perfect! The more problems you solve, the better you'll get at recognizing patterns and applying the right formulas. Keep up the great work, and don't be afraid to ask questions! Mathematics is all about exploration and discovery. This is a pretty cool result, right? If we look back at the options provided, the correct answer is (A) 1.

Summary of Key Concepts

• Exponents: Understanding how exponents work, especially the relationship between a number and its powers (e.g., aˣ). Also, the rules of exponents such as when multiplying terms with the same base, you add the exponents.
• Logarithms: Knowing the relationship between logarithms and exponents (logₐ(b) = c is the same as aᶜ = b). We used the logarithm properties to simplify equations and solve for unknown values.
• Problem-solving: Breaking down a complex problem into smaller, more manageable steps. By relating the information given to the quantities to solve for, we can simplify the problem and find the answer.

This whole process might seem a little intimidating, but breaking it down, step by step, will always make the problem more manageable. Keep practicing, and you'll become a logarithm and exponent expert in no time! Keep up the great work, and let me know if you need any more help. Happy solving, guys!