Express Log Base 2 Of (100x^3 Divided By 8) In Terms Of A And B

In this article, we will delve into the realm of logarithms and explore how to express a complex logarithmic expression in terms of simpler variables. Specifically, we are given the logarithmic relationships ${\log_x 2 = a}$ and ${\log_x 5 = b}$, and our objective is to express ${\log_2\left(\frac{100x^3}{8}\right)}$ in terms of a and b. This task requires a solid understanding of logarithmic properties, including the change of base formula, the product rule, the quotient rule, and the power rule. By strategically applying these properties, we can break down the complex expression into manageable components and ultimately express it in the desired form. This exercise not only reinforces our understanding of logarithms but also highlights their utility in simplifying complex mathematical expressions.

Before we jump into the solution, let's take a moment to review the key logarithmic properties that will be instrumental in solving this problem. A strong grasp of these properties is essential for manipulating logarithmic expressions effectively.

1. Change of Base Formula: This formula allows us to convert logarithms from one base to another. It states that ${\log_b a = \frac{\log_c a}{\log_c b}}$, where a, b, and c are positive numbers and b and c are not equal to 1. This formula is particularly useful when we need to express logarithms in a common base.
2. Product Rule: The product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. Mathematically, this is expressed as ${\log_b (mn) = \log_b m + \log_b n}$.
3. Quotient Rule: The quotient rule states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. This is expressed as ${\log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n}$.
4. Power Rule: The power rule states that the logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number. Mathematically, this is written as ${\log_b (m^p) = p \log_b m}$.

With these properties in mind, we are well-equipped to tackle the problem at hand.

Now, let's dive into the step-by-step solution to express ${\log_2\left(\frac{100x^3}{8}\right)}$ in terms of a and b. We will systematically apply the logarithmic properties discussed earlier to simplify the expression.

Step 1: Applying the Quotient Rule

Our starting expression is ${\log_2\left(\frac{100x^3}{8}\right)}$. The first step is to apply the quotient rule, which allows us to separate the logarithm of the fraction into the difference of two logarithms: ${\log_2\left(\frac{100x^3}{8}\right) = \log_2(100x^3) - \log_2(8)}$

This step simplifies the expression by breaking it down into smaller, more manageable parts. We now have two separate logarithmic terms to deal with.

Step 2: Applying the Product Rule

Next, we focus on the term ${\log_2(100x^3)}$. We can apply the product rule here, as 100 and ${x^3}$ are multiplied together: ${\log_2(100x^3) = \log_2(100) + \log_2(x^3)}$

This step further simplifies the expression by separating the product into a sum of logarithms.

Step 3: Applying the Power Rule

Now, let's address the term ${\log_2(x^3)}$. We can use the power rule to bring the exponent down as a coefficient: ${\log_2(x^3) = 3 \log_2(x)}$

This step simplifies the expression by eliminating the exponent within the logarithm.

Step 4: Simplifying ${\log_2(100)}$ and ${\log_2(8)}$

We can simplify ${\log_2(100)}$ by expressing 100 as a product of its prime factors. Since ${100 = 2^2 \cdot 5^2}$, we have: ${\log_2(100) = \log_2(2^2 \cdot 5^2)}$

Applying the product rule again: ${\log_2(100) = \log_2(2^2) + \log_2(5^2)}$

Using the power rule: ${\log_2(100) = 2 \log_2(2) + 2 \log_2(5)}$

Since ${\log_2(2) = 1}$, we get: ${\log_2(100) = 2 + 2 \log_2(5)}$

For ${\log_2(8)}$, we recognize that ${8 = 2^3}$, so: ${\log_2(8) = \log_2(2^3) = 3 \log_2(2) = 3}$

Step 5: Putting it all together

Now, let's substitute the simplified expressions back into our original equation: ${\log_2\left(\frac{100x^3}{8}\right) = \log_2(100x^3) - \log_2(8)}$ ${= \log_2(100) + \log_2(x^3) - 3}$ ${= 2 + 2 \log_2(5) + 3 \log_2(x) - 3}$ ${= 2 \log_2(5) + 3 \log_2(x) - 1}$

Step 6: Using the Change of Base Formula

We are given ${\log_x 2 = a}$ and ${\log_x 5 = b}$. We need to express ${\log_2(5)}$ and ${\log_2(x)}$ in terms of a and b. Using the change of base formula, we have: ${\log_2(x) = \frac{\log_x(x)}{\log_x(2)} = \frac{1}{a}}$

Similarly, for ${\log_2(5)}$: ${\log_2(5) = \frac{\log_x(5)}{\log_x(2)} = \frac{b}{a}}$

Step 7: Final Substitution

Substitute these expressions back into our equation: ${\log_2\left(\frac{100x^3}{8}\right) = 2 \left(\frac{b}{a}\right) + 3 \left(\frac{1}{a}\right) - 1}$ ${= \frac{2b}{a} + \frac{3}{a} - 1}$ ${= \frac{2b + 3}{a} - 1}$

Therefore, we have successfully expressed ${\log_2\left(\frac{100x^3}{8}\right)}$ in terms of a and b.

The final answer is: ${\log _2\left(\frac{100 x^3}{8}\right) = \frac{2b + 3}{a} - 1}$

This expression represents the simplified form of the original logarithm in terms of the given variables a and b. The step-by-step approach, utilizing key logarithmic properties, allowed us to break down the complex expression and arrive at the solution. Understanding and applying these properties is crucial for solving logarithmic problems efficiently and accurately.

In this comprehensive guide, we have demonstrated how to express ${\log_2\left(\frac{100x^3}{8}\right)}$ in terms of a and b, given that ${\log_x 2 = a}$ and ${\log_x 5 = b}$. By leveraging the fundamental properties of logarithms, including the quotient rule, product rule, power rule, and the change of base formula, we methodically simplified the expression. Each step was carefully explained to provide a clear understanding of the process. This exercise underscores the importance of mastering logarithmic properties for effectively manipulating and simplifying complex logarithmic expressions.

We began by outlining the core logarithmic properties necessary for solving the problem. We then applied the quotient rule to separate the fraction within the logarithm, followed by the product rule to break down the terms further. The power rule was instrumental in handling exponents within the logarithms. We simplified numerical logarithms by expressing them in terms of their prime factors and applying logarithmic properties. Finally, we used the change of base formula to express logarithms in terms of a and b, ultimately arriving at the solution: ${\frac{2b + 3}{a} - 1}$.

This problem serves as an excellent illustration of how a systematic approach, combined with a solid understanding of logarithmic principles, can lead to the successful resolution of complex mathematical problems. Whether you're a student learning about logarithms or a math enthusiast looking to sharpen your skills, this guide provides a valuable resource for mastering logarithmic manipulations. The ability to simplify and express logarithmic expressions in different forms is a fundamental skill in mathematics, with applications spanning various fields, including engineering, physics, and computer science. By practicing and applying these techniques, you can build a strong foundation in logarithmic problem-solving.