Solving Logarithmic Expressions: A Step-by-Step Guide

Dec 3, 2025 by Alex Johnson 54 views

Understanding and solving logarithmic expressions can seem daunting at first, but with a clear, step-by-step approach, you can master these problems. In this comprehensive guide, we will dissect the expression $5 \log _{10} a+\log _{20} 20-\log _{10} 10$ and identify the correct answers from the given options. We'll cover the fundamental properties of logarithms, demonstrate how to apply these properties, and walk through the solution process in a way that's easy to understand. Whether you're a student tackling homework or someone looking to refresh their math skills, this guide will provide you with the tools and knowledge you need.

Understanding the Basics of Logarithms

Before diving into the specific problem, let's recap the basic properties of logarithms. Logarithms are essentially the inverse operation to exponentiation. The logarithm of a number to a given base is the exponent to which the base must be raised to produce that number. Mathematically, if $b^y = x$ , then $\log_b x = y$ . Here, $b$ is the base, $x$ is the argument, and $y$ is the logarithm.

Key properties of logarithms that we will use in solving the expression include:

Product Rule: $\log_b(mn) = \log_b m + \log_b n$ – The logarithm of a product is the sum of the logarithms.
Quotient Rule: $\log_b(\frac{m}{n}) = \log_b m - \log_b n$ – The logarithm of a quotient is the difference of the logarithms.
Power Rule: $\log_b(m^p) = p \log_b m$ – The logarithm of a number raised to a power is the product of the power and the logarithm of the number.
Change of Base Rule: $\log_b a = \frac{\log_c a}{\log_c b}$ – This rule allows us to change the base of a logarithm.
Logarithm of Base: $\log_b b = 1$ – The logarithm of a number to the same base is 1.
Logarithm of 1: $\log_b 1 = 0$ – The logarithm of 1 to any base is 0.

With these properties in mind, we can approach the given expression systematically. Understanding these rules is crucial for simplifying and solving logarithmic equations effectively. Each rule serves as a tool in our mathematical toolbox, allowing us to manipulate and simplify complex expressions into more manageable forms.

Deconstructing the Expression: $5 \log _{10} a+\log _{20} 20-\log _{10} 10$

The given expression is $5 \log _{10} a+\log _{20} 20-\log _{10} 10$ . Let’s break it down step by step to understand each component and simplify it using the logarithmic properties we discussed.

Term 1: $5 \log _{10} a$

This term involves the logarithm of $a$ to the base 10, multiplied by 5. We can apply the power rule of logarithms here, which states that $\log_b(m^p) = p \log_b m$ . Thus, $5 \log _{10} a$ can be rewritten as $\log _{10} (a^5)$ . This transformation is a direct application of the power rule and simplifies the term by moving the coefficient into the exponent.
Term 2: $\log _{20} 20$

This is a straightforward application of the logarithmic identity $\log_b b = 1$ . The logarithm of a number to the same base is always 1. Therefore, $\log _{20} 20 = 1$ . This simplification is based on a fundamental property of logarithms and provides a constant value for this term.
Term 3: $-\log _{10} 10$

Similar to the second term, this term also uses the logarithmic identity $\log_b b = 1$ . Since the base and the argument are both 10, $\log _{10} 10 = 1$ . Therefore, $-\log _{10} 10 = -1$ . This term simplifies to a constant value due to the basic logarithmic identity.

By breaking down the expression into these three terms, we can see how each part can be simplified using fundamental logarithmic properties. The next step is to combine these simplified terms to find the overall simplified expression. This step-by-step approach makes it easier to tackle complex logarithmic problems and ensures accuracy in the simplification process.

Simplifying the Expression Step-by-Step

Now that we've deconstructed the expression $5 \log _{10} a+\log _{20} 20-\log _{10} 10$ , let’s simplify it by combining the simplified terms we found earlier. This process involves substituting the simplified forms back into the original expression and using logarithmic properties to further consolidate the terms.

Substitute the simplified terms: We found that $5 \log _{10} a = \log _{10} (a^5)$ , $\log _{20} 20 = 1$ , and $-\log _{10} 10 = -1$ . Substituting these back into the original expression gives us:

$\log _{10} (a^5) + 1 - 1$
Combine the constants: The constants $+1$ and $-1$ cancel each other out, simplifying the expression to:

$\log _{10} (a^5)$

So, the simplified form of the expression is $\log _{10} (a^5)$ . This simplification makes it easier to compare with the given options and identify the correct answers. The process of substitution and simplification is a fundamental technique in solving mathematical problems, especially those involving logarithms.

Evaluating the Given Options

Now that we have simplified the original expression to $\log_{10}(a^5)$ , let's evaluate the given options to determine which ones are correct. This involves comparing our simplified expression with each option and using logarithmic properties to check for equivalence. Each option presents a different form of a logarithmic expression, and we need to verify if they match our simplified result.

Option A: $\log _{10}(2 a)^5$

To evaluate this option, we can apply the power rule in reverse and see if it matches our simplified expression. The expression $\log _{10}(2 a)^5$ can be rewritten using the power rule as $5 \log_{10}(2a)$ . However, this does not directly match our simplified expression $\log_{10}(a^5)$ .

To further analyze, we can expand $5 \log_{10}(2a)$ using the product rule: $5(\log_{10}2 + \log_{10}a) = 5\log_{10}2 + 5\log_{10}a$ . This expression is different from $\log_{10}(a^5)$ , so Option A is incorrect.

Option B: $\log (2 a^5)$

This option presents a product inside the logarithm. Our simplified expression is $\log_{10}(a^5)$ . If we consider the base 10 logarithm, this option can be written as $\log_{10}(2a^5)$ . This expression cannot be simplified further to match $\log_{10}(a^5)$ , as there is an additional factor of 2 inside the logarithm. Therefore, Option B is incorrect.

Option C: $\log _n(100 n)+1$

This option involves a logarithm with a generic base $n$ . We can rewrite this expression using the product rule: $\log _n(100 n) = \log _n 100 + \log _n n$ . Since $\log _n n = 1$ , the expression becomes $\log _n 100 + 1$ . Adding the extra $+1$ from the original option gives us $\log _n 100 + 1 + 1 = \log _n 100 + 2$ . This expression does not resemble our simplified form $\log_{10}(a^5)$ , so Option C is incorrect.

Option D: $\log _{10}(a^5)$

This option perfectly matches our simplified expression. By recognizing the equivalence, we can confidently say that Option D is correct. This direct match confirms that our simplification process has led us to the correct form of the expression.

Conclusion

In summary, by simplifying the expression $5 \log _{10} a+\log _{20} 20-\log _{10} 10$ , we arrived at $\log_{10}(a^5)$ . Evaluating the given options, we found that only Option D, $\log _{10}(a^5)$ , is equivalent to our simplified expression. Therefore, the correct answer is D.

This exercise highlights the importance of understanding and applying logarithmic properties to simplify complex expressions. By breaking down the problem into manageable steps and systematically applying the rules, we can confidently solve logarithmic equations. The ability to manipulate logarithmic expressions is a valuable skill in mathematics and various fields of science and engineering.

For further exploration and a deeper understanding of logarithmic functions, consider visiting trusted resources such as Khan Academy's Logarithm section. This will provide you with additional practice problems and insights into the world of logarithms. Remember, practice is key to mastering these concepts, and continuous learning will enhance your mathematical skills.