Evaluating Logarithmic Expressions A Step By Step Solution

In this comprehensive guide, we will evaluate the logarithmic expression $\frac{\log 25 - \log 125 + \frac{1}{2} \log 625}{3 \log 5}$ step by step. Logarithms, often perceived as challenging, are fundamental tools in mathematics and various scientific fields. Understanding their properties and how to manipulate them is crucial for solving complex problems. This article aims to break down the problem into manageable steps, ensuring a clear understanding of the underlying principles and techniques. Whether you're a student tackling algebra problems or someone looking to refresh their math skills, this guide will provide you with the necessary knowledge and confidence to tackle similar logarithmic expressions.

Understanding Logarithms

Before diving into the specifics of the given expression, let's first solidify our understanding of logarithms. A logarithm is essentially the inverse operation of exponentiation. In simpler terms, if we have an equation like $b^x = y$, then the logarithm of $y$ to the base $b$ is $x$, written as $\log_b y = x$. This means the logarithm tells us what power we need to raise the base $b$ to in order to get the number $y$. For instance, $\log_{10} 100 = 2$ because $10^2 = 100$. Understanding this basic relationship between exponents and logarithms is the bedrock upon which all logarithmic manipulations are built. There are two commonly used bases for logarithms: base 10 (common logarithm) and base $e$ (natural logarithm). When the base is not explicitly written, it is generally assumed to be base 10. Natural logarithms, denoted as $\ln$, use the base $e$, which is approximately 2.71828. The properties we will discuss apply to logarithms of any base, but for this particular problem, we will be dealing with common logarithms (base 10).

Key Properties of Logarithms

To effectively simplify and solve logarithmic expressions, it's crucial to be well-versed in the fundamental properties of logarithms. These properties act as tools that allow us to manipulate logarithmic expressions into more manageable forms. Here, we will delve into three core properties: the product rule, the quotient rule, and the power rule. These rules are not just abstract mathematical concepts; they are the keys to unlocking the simplification of complex logarithmic problems. Mastering these properties will enable you to tackle a wide range of logarithmic challenges with confidence and precision.

1. Product Rule: The logarithm of a product is equal to the sum of the logarithms of the individual factors. Mathematically, this is expressed as $\log_b(mn) = \log_b m + \log_b n$. This rule allows us to break down complex logarithmic expressions involving multiplication into simpler addition problems. For instance, $\log(100 \times 1000)$ can be rewritten as $\log 100 + \log 1000$, making it easier to compute.

2. Quotient Rule: The logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. This can be written as $\log_b(\frac{m}{n}) = \log_b m - \log_b n$. The quotient rule is the counterpart to the product rule and allows us to handle division within logarithmic expressions. For example, $\log(\frac{1000}{10})$ can be transformed into $\log 1000 - \log 10$, simplifying the calculation.

3. Power Rule: The logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number. This is expressed as $\log_b(m^p) = p \log_b m$. The power rule is particularly useful for dealing with exponents within logarithms. For instance, $\log(10^3)$ can be simplified to $3 \log 10$, which is much easier to evaluate.

Common Logarithm Values

Before we proceed, let's take note of some common logarithm values that will be helpful in solving this problem. Recall that when the base is not explicitly written, it is assumed to be 10. Therefore, we are dealing with base-10 logarithms here. Recognizing these values beforehand can significantly speed up the simplification process. Understanding these values helps in quickly evaluating logarithmic expressions and simplifying complex equations. In the context of our problem, we will frequently encounter these values when breaking down the numbers within the expression into powers of 5, making the simplification process more streamlined and efficient.

• $\log 5$: This value will appear frequently as we simplify terms in the expression. It's essential to recognize that 25, 125, and 625 are all powers of 5.
• $\log 25 = \log (5^2) = 2 \log 5$
• $\log 125 = \log (5^3) = 3 \log 5$
• $\log 625 = \log (5^4) = 4 \log 5$

Problem Statement: Evaluate $\frac{\log 25 - \log 125 + \frac{1}{2} \log 625}{3 \log 5}$

Now that we have a solid foundation in the properties of logarithms, let's revisit the problem at hand: Evaluate the logarithmic expression $\frac{\log 25 - \log 125 + \frac{1}{2} \log 625}{3 \log 5}$. This expression appears complex at first glance, but with a systematic approach and the application of the logarithmic properties we discussed, we can simplify it effectively. The key is to break down each term, identify common factors, and apply the rules of logarithms to reduce the expression to its simplest form. By following a step-by-step method, we can transform this seemingly daunting problem into a manageable and solvable task, highlighting the power and elegance of logarithmic simplification.

Step-by-Step Solution

To solve the given logarithmic expression $\frac{\log 25 - \log 125 + \frac{1}{2} \log 625}{3 \log 5}$, we will proceed step-by-step, breaking down the problem into smaller, more manageable parts. This approach not only simplifies the calculation but also makes it easier to understand each stage of the process. We will focus on expressing each term in terms of $\log 5$ using the power rule of logarithms. This will allow us to combine like terms and simplify the expression. Let’s dive into the detailed solution:

Step 1: Express Each Term as a Power of 5

The first step in simplifying the expression is to recognize that 25, 125, and 625 are all powers of 5. This is a crucial observation because it allows us to apply the power rule of logarithms effectively. The power rule, as we discussed earlier, states that $\log_b(m^p) = p \log_b m$. By expressing each term as a power of 5, we set the stage for applying this rule and simplifying the expression. This initial transformation is a key strategy in solving many logarithmic problems, as it helps in reducing complex terms to simpler multiples of a common logarithm. This step will lay the groundwork for the subsequent simplifications and ultimately lead us to the solution.

• $\log 25 = \log (5^2)$
• $\log 125 = \log (5^3)$
• $\log 625 = \log (5^4)$

Step 2: Apply the Power Rule

Now that we have expressed the terms inside the logarithms as powers of 5, we can apply the power rule of logarithms. This rule, $\log_b(m^p) = p \log_b m$, allows us to bring the exponents outside the logarithm as coefficients. Applying the power rule is a pivotal step in simplifying logarithmic expressions, as it transforms exponents within logarithms into multiplication outside the logarithms, which is often easier to handle. In our specific problem, applying the power rule will convert the expressions into terms that are multiples of $\log 5$, making it easier to combine and simplify them further. This step is a prime example of how logarithmic properties can be used to reduce complex expressions to simpler forms.

• $\log (5^2) = 2 \log 5$
• $\log (5^3) = 3 \log 5$
• $\log (5^4) = 4 \log 5$

Step 3: Substitute Back into the Original Expression

After applying the power rule and simplifying each logarithmic term, the next step is to substitute these simplified expressions back into the original equation. This substitution is a critical step in bringing all the simplified components together into a cohesive expression. By replacing the original logarithmic terms with their simplified equivalents, we transform the initial complex equation into a more manageable form. This step allows us to see the entire expression in terms of a common logarithm, $\log 5$, which is crucial for the subsequent simplification and combination of terms. The substitution effectively bridges the gap between the initial problem and its simplified version, paving the way for the final solution.

$\frac{\log 25 - \log 125 + \frac{1}{2} \log 625}{3 \log 5} = \frac{2 \log 5 - 3 \log 5 + \frac{1}{2} (4 \log 5)}{3 \log 5}$

Step 4: Simplify the Numerator

The next critical step in solving the logarithmic expression is to simplify the numerator. This involves performing the arithmetic operations on the logarithmic terms in the numerator. By combining like terms, we can reduce the numerator to a single logarithmic expression, which will make the overall fraction much simpler to handle. This simplification typically involves basic arithmetic operations such as addition, subtraction, and multiplication. Focusing on the numerator first allows us to manage the complexity of the expression in a step-by-step manner, ensuring accuracy and clarity in the simplification process. This step is a crucial part of the overall solution, as it sets up the final division and ultimately leads to the answer.

$2 \log 5 - 3 \log 5 + \frac{1}{2} (4 \log 5) = 2 \log 5 - 3 \log 5 + 2 \log 5$

Combine the terms:

$2 \log 5 - 3 \log 5 + 2 \log 5 = (2 - 3 + 2) \log 5 = 1 \log 5 = \log 5$

Step 5: Divide by the Denominator

After simplifying the numerator, the final step is to divide the simplified numerator by the denominator. This step completes the process of reducing the entire expression to its simplest form. In our case, the simplified numerator is $\log 5$, and the denominator is $3 \log 5$. Dividing the numerator by the denominator involves basic algebraic division, where we look for common factors that can be canceled out. This division is a straightforward process that leads directly to the final answer. It highlights how the previous steps of simplification have streamlined the problem, making the final calculation simple and direct.

$\frac{\log 5}{3 \log 5}$

Notice that $\log 5$ appears in both the numerator and the denominator, so we can cancel it out:

$\frac{\log 5}{3 \log 5} = \frac{1}{3}$

Final Answer

Therefore, after carefully applying the properties of logarithms and simplifying step by step, we have successfully evaluated the logarithmic expression $\frac{\log 25 - \log 125 + \frac{1}{2} \log 625}{3 \log 5}$. The final answer is $\frac{1}{3}$. This result demonstrates the power and elegance of using logarithmic properties to simplify complex expressions. By breaking down the problem into manageable steps and applying the appropriate rules, we were able to transform a seemingly daunting expression into a simple fraction. This process highlights the importance of a systematic approach in mathematics, where understanding the underlying principles and applying them methodically can lead to a clear and concise solution.

Conclusion

In conclusion, we have successfully evaluated the logarithmic expression $\frac{\log 25 - \log 125 + \frac{1}{2} \log 625}{3 \log 5}$ by systematically applying the fundamental properties of logarithms. The step-by-step approach we followed involved expressing each term as a power of 5, applying the power rule, substituting the simplified terms back into the original expression, simplifying the numerator, and finally, dividing by the denominator. This process not only yielded the final answer of $\frac{1}{3}$ but also provided a clear and understandable pathway for solving similar logarithmic problems. Understanding and mastering these logarithmic properties is crucial for anyone studying mathematics, as they are frequently used in various branches of the field, including algebra, calculus, and trigonometry. This comprehensive guide serves as a valuable resource for students, educators, and anyone seeking to enhance their understanding of logarithms and their applications. By breaking down the problem into manageable steps, we have demonstrated that even complex logarithmic expressions can be solved with the right knowledge and approach.