Simplifying The Expression (π Log 125)(log₁₀³⁶) / (log₁₀²¹) (log₁₀⁶²⁵) A Math Guide

Hey guys! Let's dive into simplifying this math expression together. We're tackling $\frac{(\pi \log 125)(\log _{10}^{36})}{(\log _{10}^{21})(\log _{10}^{625})}$, and it might look a bit intimidating at first, but don't worry, we'll break it down step by step. Remember, the key to simplifying complex expressions is to understand the underlying mathematical principles and apply them methodically. We'll use properties of logarithms, prime factorization, and some good old-fashioned algebraic manipulation to get to the simplest form. So, grab your calculators (or not, if you're feeling like a math whiz today!) and let's get started. We will explore each component of the expression individually before combining them. This approach will make the entire process easier to understand and manage. Let's begin by focusing on the logarithmic terms and see how we can simplify them using logarithmic properties.

Understanding the Components

Before we jump into the solution, let’s understand what we’re working with. The expression has logarithms with different bases and arguments, and a constant π.

• π (pi): This is the famous mathematical constant, approximately equal to 3.14159.
• log 125: This is the logarithm of 125, and we'll aim to express 125 as a power of a prime number to simplify it.
• log₁₀³⁶: This represents the base-10 logarithm of 36. We will simplify this by expressing 36 as a product of its prime factors.
• log₁₀²¹: Similarly, this is the base-10 logarithm of 21, which we’ll simplify using prime factorization.
• log₁₀⁶²⁵: This is the base-10 logarithm of 625. Like 125, we'll express 625 as a power of a prime number.

Now that we have a handle on the individual components, we can move on to breaking down each logarithmic term. This involves using prime factorization and logarithmic identities to express the terms in a simpler form. By understanding the composition of each number within the logarithms, we can apply the power rule and product rule of logarithms to simplify the expression. This is where the real magic happens, guys! So, let's roll up our sleeves and dive into the nitty-gritty details.

Breaking Down the Logarithmic Terms

Let's simplify each logarithmic term individually:

Simplifying log 125

First, we express 125 as a power of its prime factor. 125 is 5³, so we have:

log 125 = log (5³)

Using the power rule of logarithms (logₐ(bⁿ) = n logₐ(b)), we get:

log (5³) = 3 log 5

Simplifying log₁₀³⁶

We express 36 as a product of its prime factors: 36 = 2² * 3². So,

log₁₀ 36 = log₁₀ (2² * 3²)

Using the product rule of logarithms (logₐ(mn) = logₐ(m) + logₐ(n)), we get:

log₁₀ (2² * 3²) = log₁₀ (2²) + log₁₀ (3²)

Now, using the power rule, we have:

log₁₀ (2²) + log₁₀ (3²) = 2 log₁₀ 2 + 2 log₁₀ 3

Simplifying log₁₀²¹

Expressing 21 as a product of its prime factors, 21 = 3 * 7. Thus,

log₁₀ 21 = log₁₀ (3 * 7)

Applying the product rule of logarithms:

log₁₀ (3 * 7) = log₁₀ 3 + log₁₀ 7

Simplifying log₁₀⁶²⁵

Express 625 as a power of its prime factor. 625 is 5⁴, so:

log₁₀ 625 = log₁₀ (5⁴)

Using the power rule of logarithms:

log₁₀ (5⁴) = 4 log₁₀ 5

By simplifying each logarithmic term individually, we've made the original expression much more manageable. This step-by-step approach allows us to see the structure of the problem more clearly. Next, we'll substitute these simplified terms back into the original expression. This will give us a new expression that is easier to work with, as we've eliminated the composite numbers within the logarithms. This is like organizing your tools before starting a big project – it sets you up for success. Now, let's put these pieces back together and see what we get!

Substituting the Simplified Terms

Now that we've simplified each logarithmic term, let's substitute them back into the original expression:

Original expression:

$\frac{(\pi \log 125)(\log _{10}^{36})}{(\log _{10}^{21})(\log _{10}^{625})}$

Substituting the simplified terms:

$\frac{(\pi imes 3 \log 5)(2 \log _{10} 2 + 2 \log _{10} 3)}{((\log _{10} 3 + \log _{10} 7)(4 \log _{10} 5))}$

Now, let’s rearrange the terms to make the simplification process clearer:

$\frac{3\pi \log 5 imes 2(\log _{10} 2 + \log _{10} 3)}{4 \log _{10} 5 (\log _{10} 3 + \log _{10} 7)}$

This looks much cleaner, doesn't it? We can already see some terms that can be cancelled out. The next step is to cancel out common factors. This involves identifying terms in the numerator and the denominator that are the same, allowing us to simplify the expression further. Think of it as decluttering – getting rid of the unnecessary bits to reveal the simpler form underneath. By canceling common factors, we make the expression easier to handle and closer to its final simplified state. Let's dive into the cancellation process and see what we can eliminate!

Cancelling Common Factors

We can cancel out common factors from the numerator and the denominator. We have log 5 in both the numerator and the denominator, so we can cancel those out:

$\frac{3\pi \cancel{\log 5} imes 2(\log _{10} 2 + \log _{10} 3)}{4 \cancel{\log _{10} 5} (\log _{10} 3 + \log _{10} 7)}$

This simplifies the expression to:

$\frac{6\pi(\log _{10} 2 + \log _{10} 3)}{4(\log _{10} 3 + \log _{10} 7)}$

We can further simplify the fraction by dividing both the numerator and the denominator by 2:

$\frac{3\pi(\log _{10} 2 + \log _{10} 3)}{2(\log _{10} 3 + \log _{10} 7)}$

Now, let's pause here. We've canceled out the common factors and simplified the expression as much as we can using basic algebraic manipulations. At this point, we need to evaluate whether we can simplify further or if this is the simplest form. The remaining logarithmic terms don't have obvious simplifications, and we don't have numerical values to substitute directly. So, let's take a closer look at what we have and decide on our next steps. Sometimes, recognizing when you've reached the simplest form is just as important as the simplification process itself. So, let's reflect on our progress and determine the final answer.

Final Simplified Expression

After canceling the common factors and simplifying, we've arrived at:

$\frac{3\pi(\log _{10} 2 + \log _{10} 3)}{2(\log _{10} 3 + \log _{10} 7)}$

Let's think about whether we can simplify this further. We have logarithms of prime numbers (2, 3, and 7), and there are no obvious ways to combine these terms to simplify the expression significantly. So, this might be the simplest form we can achieve without using approximations or numerical evaluations.

Therefore, the simplified expression is:

$\frac{3\pi(\log _{10} 2 + \log _{10} 3)}{2(\log _{10} 3 + \log _{10} 7)}$

This is our final answer, guys! We took a complex-looking expression and broke it down step by step, using the properties of logarithms and prime factorization. It might seem like a lot of work, but each step was logical and built upon the previous one. Remember, when you're faced with a challenging math problem, breaking it down into smaller, manageable parts is the key to success. You've nailed it! Give yourself a pat on the back for making it through this simplification journey. And remember, the more you practice, the more comfortable you'll become with these kinds of problems. Keep up the awesome work!

Conclusion

In this guide, we successfully simplified the given mathematical expression by breaking it down into manageable steps. We utilized prime factorization and the properties of logarithms, such as the power rule and product rule, to simplify each term. By methodically canceling common factors, we arrived at the simplified form:

$\frac{3\pi(\log _{10} 2 + \log _{10} 3)}{2(\log _{10} 3 + \log _{10} 7)}$

This process underscores the importance of understanding fundamental mathematical principles and applying them systematically. Remember, guys, every complex problem can be solved by breaking it down into simpler steps. Keep practicing, and you'll become a math whiz in no time!