Equivalent Logarithmic Expressions: A Step-by-Step Guide

Have you ever stumbled upon a logarithmic expression and wondered if there's a simpler way to represent it? You're not alone! Logarithms can seem daunting, but with a few key properties and a bit of practice, you can master the art of simplifying and finding equivalent expressions. In this article, we'll break down a specific logarithmic expression, $5 \log _{10} x+\log _{10} 20-\log _{10} 10$, and explore how to identify equivalent forms. Let's dive in and unlock the secrets of logarithmic expressions!

Understanding the Core Concepts of Logarithms

Before we jump into the problem, let's refresh our understanding of logarithms. At its heart, a logarithm answers the question: "To what power must we raise the base to get a certain number?" In the expression $\log_{b} a = c$, b is the base, a is the argument, and c is the exponent. This means that $b^c = a$. When the base isn't explicitly written, like in our expression with $\log_{10}$, it's understood to be base 10, also known as the common logarithm. Understanding this foundational concept is crucial for manipulating and simplifying logarithmic expressions. This understanding forms the basis for applying logarithmic properties effectively.

Furthermore, understanding the properties of logarithms is vital. These properties are the tools we'll use to manipulate and simplify expressions. The key properties include:

• Product Rule: $\log_{b}(mn) = \log_{b}m + \log_{b}n$
• Quotient Rule: $\log_{b}(m/n) = \log_{b}m - \log_{b}n$
• Power Rule: $\log_{b}(m^p) = p \log_{b}m$

These properties allow us to combine, separate, and simplify logarithmic expressions. For example, the power rule is especially useful for dealing with exponents within logarithms, and the product and quotient rules help us manage multiplication and division. Keeping these rules in mind, we can approach the given expression with a clear strategy for simplification. Think of these properties as the fundamental building blocks for solving logarithmic problems, allowing us to transform complex expressions into simpler, equivalent forms.

Breaking Down the Given Expression: $5 \log _{10} x+\log _{10} 20-\log _{10} 10$

Our mission is to find expressions equivalent to $5 \log _{10} x+\log _{10} 20-\log _{10} 10$. To do this effectively, we'll leverage the properties of logarithms we discussed earlier. Let's tackle this step by step, applying the rules in a strategic manner to simplify the expression. We'll start by focusing on the power rule, then move to combining terms using the product and quotient rules. By carefully applying each property, we can transform the original expression into a more manageable form, making it easier to identify equivalent expressions. The key is to break down the problem into smaller, more digestible steps, ensuring we don't miss any opportunities for simplification.

First, we'll apply the power rule to the term $5 \log_{10} x$. The power rule states that $\log_{b}(m^p) = p \log_{b}m$. Applying this in reverse, we can rewrite $5 \log_{10} x$ as $\log_{10} (x^5)$. This step is crucial because it consolidates the coefficient into the argument of the logarithm, setting the stage for further simplification. By moving the coefficient, we make the expression more amenable to the product and quotient rules, which will allow us to combine the logarithmic terms.

Now our expression looks like this: $\log_{10} (x^5) + \log_{10} 20 - \log_{10} 10$. Next, we'll use the product rule to combine the first two terms. The product rule states that $\log_{b}(mn) = \log_{b}m + \log_{b}n$. Applying this, we can combine $\log_{10} (x^5) + \log_{10} 20$ into $\log_{10} (20x^5)$. This step significantly simplifies the expression by merging two logarithmic terms into one, making it easier to handle the subtraction that follows. By using the product rule, we're essentially reversing the process of separating logarithms, bringing us closer to a single, simplified logarithmic expression.

Our expression is now: $\log_{10} (20x^5) - \log_{10} 10$. Finally, we'll apply the quotient rule to combine the remaining terms. The quotient rule states that $\log_{b}(m/n) = \log_{b}m - \log_{b}n$. Applying this, we can rewrite $\log_{10} (20x^5) - \log_{10} 10$ as $\log_{10} \frac{20x^5}{10}$. This step is the final key to simplifying the expression, as it combines the subtraction of logarithms into a single logarithm of a quotient. The result is a more compact form that is easier to analyze and compare with other expressions.

Simplifying the fraction inside the logarithm, we get $\log_{10} (2x^5)$. This is our simplified expression. By carefully applying the power, product, and quotient rules, we've transformed the original expression into a much cleaner form. This final form is not only easier to understand but also makes it straightforward to identify equivalent expressions. The process of simplification highlights the power of logarithmic properties in making complex expressions more manageable.

Identifying Equivalent Expressions

Now that we've simplified the original expression to $\log_{10} (2x^5)$, let's examine the given options to see which ones are equivalent. This involves further manipulation using logarithmic properties and algebraic simplification. We'll compare each option with our simplified expression to determine if they are indeed equivalent. This step is crucial in solidifying our understanding of logarithmic properties and their application in identifying equivalent expressions.

Option A: $\log _{10}\left(2 x^5\right)$

This option is exactly the simplified form we derived: $\log_{10} (2x^5)$. Therefore, Option A is equivalent to the original expression. This direct match confirms the effectiveness of our simplification process and provides a clear benchmark for evaluating the other options.

Option B: $\log _{10}(100 x)+1$

To determine if this is equivalent, let's manipulate it. We can rewrite the constant 1 as $\log_{10} 10$. So the expression becomes $\log_{10}(100x) + \log_{10} 10$. Applying the product rule, we get $\log_{10}(100x imes 10) = \log_{10}(1000x)$. This is not equivalent to $\log_{10} (2x^5)$. The difference in the argument of the logarithm clearly indicates that these two expressions are not the same. This highlights the importance of careful manipulation and comparison when dealing with logarithmic expressions.

Option C: $\log _{10}(10 x)$

This expression is significantly simpler than our simplified form, $\log_{10} (2x^5)$. There's no way to manipulate $\log_{10}(10x)$ to get $\log_{10} (2x^5)$, so Option C is not equivalent. The absence of the exponent on x and the different constant factor make it clear that these expressions are distinct.

Option D: $\log _{10}(2$

This option seems incomplete and doesn't have the variable x, making it not equivalent to the original expression. Without the variable x, this expression cannot represent the same relationship as the original, which clearly depends on the value of x.

Conclusion

In this comprehensive guide, we've successfully navigated the world of logarithmic expressions. We started with the expression $5 \log _{10} x+\log _{10} 20-\log {10} 10$, and through the strategic application of logarithmic properties—the power rule, product rule, and quotient rule—we simplified it to $\log{10} (2x^5)$. This step-by-step simplification not only made the expression easier to understand but also allowed us to confidently identify equivalent expressions.

We then evaluated four options, meticulously comparing each one to our simplified form. Our analysis revealed that only _Option A, $\log {10}\left(2 x^5\right)$, is equivalent to the original expression. This exercise underscored the importance of mastering logarithmic properties and applying them systematically to manipulate and simplify expressions.

Understanding and applying these properties is crucial for anyone working with logarithms, whether in mathematics, science, or engineering. By breaking down complex expressions into simpler forms, we can gain deeper insights and solve problems more effectively. Remember, practice is key! The more you work with logarithmic expressions, the more comfortable and confident you'll become in manipulating them.

To further enhance your understanding of logarithms, explore resources like Khan Academy's Logarithm section, which offers a wealth of tutorials, practice problems, and videos. Keep exploring, keep practicing, and you'll master the art of logarithms in no time!