Solving Logarithmic Equations Finding The Value Of Logarithmic Expressions

In the realm of mathematics, logarithms serve as a powerful tool for simplifying complex calculations and unraveling intricate relationships between numbers. Logarithmic expressions provide a concise way to represent exponents, offering a unique perspective on mathematical operations. In this article, we will delve into the world of logarithmic expressions, exploring their properties and applications, and ultimately unraveling the solution to a specific logarithmic problem.

Deciphering Logarithmic Notation

At its core, a logarithm is the inverse operation of exponentiation. In simpler terms, the logarithm of a number to a given base is the exponent to which the base must be raised to produce that number. The notation for a logarithm is as follows:

$$\log_b A = x$$

where:

• b is the base of the logarithm
• A is the argument of the logarithm (the number whose logarithm we are finding)
• x is the logarithm of A to the base b (the exponent to which b must be raised to equal A)

For instance, the expression $\log_2 8 = 3$ signifies that the logarithm of 8 to the base 2 is 3, because 2 raised to the power of 3 equals 8 (2³ = 8). Understanding this fundamental relationship between logarithms and exponents is crucial for manipulating logarithmic expressions effectively.

Key Properties of Logarithms

Logarithms possess several key properties that enable us to simplify and manipulate logarithmic expressions. These properties are derived from the fundamental relationship between logarithms and exponents, and they serve as indispensable tools for solving logarithmic problems.

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 property allows us to break down complex expressions involving products into simpler terms, making them easier to manage.

2. Quotient Rule: The logarithm of a quotient is equal to the difference between the logarithms of the numerator and the denominator. Mathematically, this is expressed as:

   $$\log_b (M/N) = \log_b M - \log_b N$$

   This property is particularly useful for simplifying expressions involving fractions or divisions.

3. Power Rule: The logarithm of a number raised to a power is equal to the product of the power and the logarithm of the number. Mathematically, this is expressed as:

   $$\log_b (M^p) = p \log_b M$$

   This property allows us to deal with exponents within logarithmic expressions, making them more manageable.

4. Change of Base Rule: This rule allows us to convert logarithms from one base to another. Mathematically, it is expressed as:

   $$\log_a M = \frac{\log_b M}{\log_b a}$$

   This property is particularly useful when dealing with logarithms of different bases, as it allows us to express them in a common base for easier comparison or calculation.

By mastering these properties, we equip ourselves with the necessary tools to tackle a wide range of logarithmic problems.

Solving the Logarithmic Problem

Now, let's apply our understanding of logarithmic properties to solve the problem presented in the prompt. We are given the following information:

$$\log_b A = 3$$

$$\log_b C = 2$$

$$\log_b D = 5$$

Our goal is to find the value of the expression:

$$\log_b\left(\frac{A^5 C^2}{D^6}\right)$$

To achieve this, we will leverage the properties of logarithms to break down the expression into simpler terms. First, we apply the quotient rule:

$$\log_b\left(\frac{A^5 C^2}{D^6}\right) = \log_b (A^5 C^2) - \log_b (D^6)$$

Next, we apply the product rule to the first term:

$$\log_b (A^5 C^2) = \log_b (A^5) + \log_b (C^2)$$

Now, we apply the power rule to each term:

$$\log_b (A^5) = 5 \log_b A$$

$$\log_b (C^2) = 2 \log_b C$$

$$\log_b (D^6) = 6 \log_b D$$

Substituting the given values for $\log_b A$, $\log_b C$, and $\log_b D$, we get:

$$5 \log_b A = 5 \times 3 = 15$$

$$2 \log_b C = 2 \times 2 = 4$$

$$6 \log_b D = 6 \times 5 = 30$$

Now, we substitute these values back into our original expression:

$$\log_b\left(\frac{A^5 C^2}{D^6}\right) = (15 + 4) - 30$$

Simplifying the expression, we get:

$$\log_b\left(\frac{A^5 C^2}{D^6}\right) = 19 - 30 = -11$$

Therefore, the value of the logarithmic expression $\log_b\left(\frac{A^5 C^2}{D^6}\right)$ is -11.

Conclusion

In this article, we have explored the world of logarithmic expressions, delving into their properties and applications. We have learned how to decipher logarithmic notation, understand the key properties of logarithms, and apply these properties to solve logarithmic problems. By mastering these concepts, we can confidently tackle a wide range of mathematical challenges involving logarithms.

The solution to the specific logarithmic problem presented in the prompt is -11, which corresponds to option D. This solution demonstrates the power of logarithmic properties in simplifying complex expressions and arriving at accurate results. As we continue our mathematical journey, the knowledge and skills gained in this exploration of logarithms will undoubtedly prove invaluable.

Logarithms are a fundamental concept in mathematics, acting as the inverse operation to exponentiation. They provide a powerful tool for simplifying calculations and solving equations involving exponents. In essence, a logarithm answers the question: