Condensing Logarithmic Expressions How To Simplify 1/8(log2 Z + 3 Log2 Y)

In mathematics, especially in dealing with logarithmic expressions, condensing or simplifying expressions is a fundamental skill. Logarithmic expressions can often appear complex, but by applying the properties of logarithms, we can reduce them to a more manageable form. This article aims to provide a detailed explanation of how to condense the given logarithmic expression using the properties of logarithms, with the goal of simplifying it into a single logarithmic term. We will walk through each step, explaining the underlying principles and properties used, to help you understand the process thoroughly. This skill is crucial not only in academic settings but also in various applications in science, engineering, and finance, where logarithmic scales and computations are frequently encountered.

Understanding Logarithmic Properties

To effectively condense logarithmic expressions, it is essential to grasp the core properties of logarithms. These properties allow us to manipulate logarithmic expressions and combine or separate logarithmic terms as needed. The main properties we will use in this article are the power rule and the product rule. Let's briefly recap these properties to ensure we have a solid foundation before diving into the problem.

1. The Power Rule: The power rule of logarithms states that the logarithm of a number raised to a power is the product of the power and the logarithm of the number. Mathematically, this is expressed as:

   $$\log_b(x^n) = n \log_b(x)$$

   Where n is any real number, b is the base of the logarithm, and x is the argument. This rule allows us to move exponents from inside the logarithm to the outside as coefficients, or vice versa. The power rule is particularly useful when dealing with terms like $3 \log_2 y$ in our given expression.

2. The Product Rule: The product rule of logarithms states that the logarithm of the product of two numbers is the sum of the logarithms of the individual numbers. Mathematically, this is expressed as:

   $$\log_b(xy) = \log_b(x) + \log_b(y)$$

   This rule enables us to combine multiple logarithmic terms into a single logarithm, which is essential for condensing expressions. It is the inverse operation of expanding a logarithm of a product into separate logarithmic terms. By applying the product rule, we can merge terms that are added together within a logarithmic expression.

Understanding these logarithmic properties is crucial for effectively simplifying and manipulating logarithmic expressions. By mastering these rules, one can confidently tackle various problems involving logarithms, including the condensation of complex expressions into simpler forms. In the following sections, we will apply these properties to the given expression, providing step-by-step instructions to achieve the condensed form.

Problem Statement

Our primary task is to condense the given expression into a single logarithmic term. The expression is:

$$\frac{1}{8}(\log_2 z + 3 \log_2 y)$$

This expression involves a fraction multiplied by a sum of logarithmic terms. To condense this, we will use the logarithmic properties we discussed earlier. The goal is to rewrite the expression in the form of a single logarithm, such as $\log_2(expression)$. This involves multiple steps, each utilizing a specific property of logarithms. First, we will address the coefficient inside the parenthesis and then the fraction outside the parenthesis.

Step-by-Step Solution

Step 1: Apply the Power Rule

The first step in condensing the expression is to deal with the coefficient of the second term inside the parentheses. We have $3 \log_2 y$. According to the power rule, $n \log_b(x) = \log_b(x^n)$. Applying this rule, we can rewrite $3 \log_2 y$ as $\log_2(y^3)$. This transforms our expression into:

$$\frac{1}{8}(\log_2 z + \log_2 y^3)$$

By using the power rule, we have eliminated the coefficient 3, making it easier to combine the terms inside the parentheses. This step is crucial because it allows us to proceed with the next step, which involves combining the logarithms using the product rule.

Step 2: Apply the Product Rule

Now that we have two logarithmic terms inside the parentheses, we can apply the product rule to combine them. The product rule states that $\log_b(x) + \log_b(y) = \log_b(xy)$. In our case, we have $\log_2 z + \log_2 y^3$. Applying the product rule, we get:

$$\log_2(z imes y^3) = \log_2(zy^3)$$

Thus, our expression becomes:

$$\frac{1}{8} \log_2(zy^3)$$

By applying the product rule, we have successfully combined the two logarithmic terms into a single term, which is a significant step towards condensing the entire expression. The argument of the logarithm is now the product of z and y cubed.

Step 3: Apply the Power Rule Again

Finally, we need to deal with the fraction $\frac{1}{8}$ outside the logarithm. We can use the power rule again, but this time in reverse. Recall that $n \log_b(x) = \log_b(x^n)$. We can rewrite $\frac{1}{8} \log_2(zy^3)$ as $\log_2((zy^3)^{\frac{1}{8}})$. This means we are raising the entire argument $zy^3$ to the power of $\frac{1}{8}$.

$$\log_2((zy^3)^{\frac{1}{8}})$$

Now, we can simplify the exponent $\frac{1}{8}$. Raising a term to the power of $\frac{1}{8}$ is the same as taking the eighth root of the term. Therefore, we can rewrite the expression as:

$$\log_2(\sqrt[8]{zy^3})$$

This is the fully condensed form of the given expression. We have successfully combined all the logarithmic terms into a single logarithm with a simplified argument.

Final Condensed Form

After applying the logarithmic properties step by step, we have condensed the expression:

$$\frac{1}{8}(\log_2 z + 3 \log_2 y)$$

into its simplest logarithmic form:

$$\log_2(\sqrt[8]{zy^3})$$

This final form represents the original expression as a single logarithm, which is often more convenient for further calculations or analysis. The process involved using both the power rule and the product rule of logarithms, demonstrating the importance of understanding and applying these properties effectively.

Conclusion

In this article, we have demonstrated how to condense a given logarithmic expression into its simplest form. We started with the expression $\frac{1}{8}(\log_2 z + 3 \log_2 y)$ and, through a series of steps, transformed it into $\log_2(\sqrt[8]{zy^3})$. This process involved a clear understanding and application of the logarithmic properties, specifically the power rule and the product rule.

The ability to condense logarithmic expressions is a valuable skill in various fields, including mathematics, physics, engineering, and computer science. It simplifies complex expressions, making them easier to work with and interpret. The step-by-step approach outlined in this article provides a clear methodology for tackling such problems.

To summarize, the key steps included:

1. Applying the Power Rule to move coefficients inside the logarithm as exponents.
2. Applying the Product Rule to combine multiple logarithmic terms into a single term.
3. Applying the Power Rule again to address the fractional coefficient outside the logarithm.

By mastering these steps and understanding the underlying logarithmic properties, you can confidently condense various logarithmic expressions and simplify complex mathematical problems. This not only enhances your problem-solving skills but also provides a deeper understanding of logarithmic functions and their applications.

Therefore, the condensed form of the given expression is:

$$\log_2(\sqrt[8]{zy^3})$$

Q: Why is it important to condense logarithmic expressions? A: Condensing logarithmic expressions simplifies complex mathematical expressions, making them easier to work with and interpret. This skill is valuable in various fields, including mathematics, physics, engineering, and computer science.

Q: What are the key properties of logarithms used in condensing expressions? A: The key properties used are the power rule ($\log_b(x^n) = n \log_b(x)$) and the product rule ($\log_b(xy) = \log_b(x) + \log_b(y)$).

Q: How does the power rule help in condensing logarithmic expressions? A: The power rule allows us to move coefficients inside the logarithm as exponents, which helps in combining terms and simplifying the expression.

Q: How does the product rule help in condensing logarithmic expressions? A: The product rule allows us to combine multiple logarithmic terms into a single term, which is a crucial step in condensing expressions.

Q: What is the condensed form of the expression $\frac{1}{8}(\log_2 z + 3 \log_2 y)$? A: The condensed form is $\log_2(\sqrt[8]{zy^3})$.