Solving Logarithmic Equations Filling In Missing Values

#h1 Fill in the Missing Values to Make the Equations True: A Comprehensive Guide

This article provides a comprehensive guide to solving logarithmic equations by filling in the missing values. We will explore the fundamental properties of logarithms and apply them to solve three different types of problems. This detailed explanation will help students and anyone interested in mathematics to understand and master the concepts of logarithms. Our exploration will cover the essential logarithmic properties, providing step-by-step solutions and insightful explanations to enhance your understanding. Let’s dive in and unravel the complexities of logarithmic equations together.

(a) ${\log _8 5-\log _8 3=\log _8 \square}$

Understanding the Logarithmic Property

To solve this equation, we need to recall the quotient rule of logarithms. The quotient rule states that the logarithm of a quotient is equal to the difference of the logarithms. Mathematically, this can be expressed as:

$$\log_b{M} - \log_b{N} = \log_b{\frac{M}{N}}$$

In this case, we have ${\log _8 5-\log _8 3}$, which fits the left side of the quotient rule. Applying the rule, we can rewrite the equation as:

$$\log _8 5-\log _8 3 = \log _8 \frac{5}{3}$$

Now, we can see that the missing value is the result of the division ${\frac{5}{3}}$. This section of the article aims to delve into the application of the quotient rule of logarithms, which is a fundamental concept in simplifying and solving logarithmic expressions. Understanding this rule is crucial for anyone looking to master logarithmic equations. The quotient rule allows us to combine two logarithms with the same base that are being subtracted into a single logarithm of a quotient. This is a powerful tool that simplifies complex expressions and makes them easier to solve. For example, consider the expression ${\log_2 16 - \log_2 4}$. According to the quotient rule, we can rewrite this as ${\log_2 \frac{16}{4}}$, which simplifies to ${\log_2 4}$. This not only simplifies the expression but also makes it easier to evaluate. In our given problem, ${\log _8 5-\log _8 3}$, we apply the same principle. We identify that we have two logarithms with the same base (8) that are being subtracted. By applying the quotient rule, we combine these into a single logarithm of the quotient of their arguments. This transforms the expression into ${\log _8 \frac{5}{3}}$. This step is essential because it directly reveals the missing value in the original equation. The quotient rule is not just a formula to memorize; it’s a tool to understand the underlying relationship between logarithmic operations and arithmetic operations. Subtraction of logarithms corresponds to division of their arguments, a relationship that stems from the properties of exponents. Logarithms are, after all, the inverse operation of exponentiation, and understanding this connection is key to mastering logarithmic manipulations. Moreover, the application of the quotient rule is not limited to simple numerical expressions. It can be used in more complex algebraic contexts where the arguments of the logarithms are variables or functions. This versatility makes the quotient rule an indispensable tool in various mathematical problems, including calculus and differential equations. By mastering this rule, students can approach logarithmic equations with confidence and clarity, knowing they have a powerful technique at their disposal.

Solution

Therefore, the missing value is ${\frac{5}{3}}$.

So, the complete equation is:

$$\log _8 5-\log _8 3=\log _8 \frac{5}{3}$$

(b) ${\log _5 9+\log _5 \square=\log _5 99}$

Applying the Product Rule of Logarithms

To solve this, we need to use the product rule of logarithms. The product rule states that the logarithm of a product is equal to the sum of the logarithms. This can be written as:

$$\log_b{MN} = \log_b{M} + \log_b{N}$$

In our equation, we have ${\log _5 9+\log _5 \square=\log _5 99}$. Let's denote the missing value as x. So, the equation becomes:

$$\log _5 9+\log _5 x=\log _5 99$$

Applying the product rule, we combine the left side:

$$\log _5 (9x) = \log _5 99$$

This section elaborates on the application of the product rule of logarithms, a cornerstone in manipulating logarithmic expressions. The product rule is the counterpart to the quotient rule, but instead of dealing with division and subtraction, it deals with multiplication and addition. Understanding this rule allows us to simplify logarithmic expressions where terms are being added. The product rule states that the sum of the logarithms of two numbers is equal to the logarithm of the product of those numbers, provided they have the same base. Mathematically, it is represented as ${\log_b{MN} = \log_b{M} + \log_b{N}}$. This rule stems directly from the properties of exponents. Since logarithms are the inverse of exponential functions, they inherit many properties from exponents. Just as multiplying exponential terms with the same base involves adding their exponents, adding logarithms with the same base corresponds to multiplying their arguments. In the context of our equation, ${\log _5 9+\log _5 x=\log _5 99}$, we have two logarithmic terms on the left side that are being added. To solve this, we apply the product rule to combine these two logarithms into a single logarithm. This is a crucial step because it simplifies the equation and sets the stage for solving for the unknown value. By applying the product rule, we transform the equation into ${\log _5 (9x) = \log _5 99}$. This equation is much simpler to deal with because we have consolidated the logarithmic terms on one side. The product rule is not just a technique for solving equations; it also provides a deeper insight into the nature of logarithms. It highlights the relationship between logarithmic and arithmetic operations, allowing us to manipulate complex expressions more effectively. Furthermore, the product rule is applicable in a wide range of contexts, from solving simple equations to more advanced problems in calculus and engineering. By mastering this rule, you gain a valuable tool that enhances your problem-solving capabilities in mathematics and beyond.

Solving for the Missing Value

Since the logarithms are equal and have the same base, their arguments must be equal:

$$9x = 99$$

Now, we solve for x:

$$x = \frac{99}{9}$$

$$x = 11$$

Solution

Therefore, the missing value is 11.

The complete equation is:

$$\log _5 9+\log _5 11=\log _5 99$$

(c) ${\log _2 25=\square \log _2 5}$

Utilizing the Power Rule of Logarithms

To solve this, we need to apply the power rule of logarithms. The power rule states that the logarithm of a number raised to a power is equal to the power times the logarithm of the number. This can be written as:

$$\log_b{M^p} = p \log_b{M}$$

In our equation, we have ${\log _2 25}$. We can rewrite 25 as ${5^2}$. So, the equation becomes:

$$\log _2 25 = \log _2 {5^2}$$

Applying the power rule, we get:

$$\log _2 {5^2} = 2 \log _2 5$$

This section provides an in-depth explanation of the power rule of logarithms, which is essential for handling logarithms of numbers raised to a power. The power rule simplifies logarithmic expressions by allowing us to move exponents from the argument of the logarithm to a coefficient in front of the logarithm. This rule is a direct consequence of the properties of exponents and their inverse relationship with logarithms. Mathematically, the power rule is expressed as ${\log_b{M^p} = p \log_b{M}}$. This means that if you have a logarithm of a number raised to a power, you can bring the exponent down and multiply it by the logarithm of the base number. This transformation can significantly simplify complex logarithmic expressions. To illustrate, consider the expression ${\log_2 8^3}$. Using the power rule, we can rewrite this as ${3 \log_2 8}$. Since we know that ${\log_2 8 = 3}$, the expression further simplifies to ${3 \times 3 = 9}$. This example showcases the power rule's ability to make calculations more manageable. In our given equation, ${\log _2 25=\square \log _2 5}$, the power rule is particularly useful. We recognize that 25 can be expressed as ${5^2}$. By rewriting 25 as ${5^2}$, we can apply the power rule to the left side of the equation. This transforms ${\log _2 25}$ into ${\log _2 {5^2}}$, which then becomes ${2 \log _2 5}$ using the power rule. This transformation is crucial because it directly reveals the missing value in the equation. The power rule is not just a formula to be memorized; it’s a tool that provides a deeper understanding of the structure of logarithms. It highlights how logarithms interact with exponents, allowing us to manipulate expressions in a way that exposes their underlying mathematical relationships. Mastery of the power rule is essential for anyone studying logarithms, as it appears frequently in various mathematical contexts, including calculus, algebra, and beyond.

Solution

Therefore, the missing value is 2.

The complete equation is:

$$\log _2 25=2 \log _2 5$$

Conclusion

In this article, we have explored how to fill in the missing values in logarithmic equations using the quotient, product, and power rules of logarithms. These rules are fundamental in simplifying and solving logarithmic expressions. By understanding and applying these rules, you can confidently tackle various logarithmic problems. Mastering these concepts not only enhances your mathematical skills but also provides a deeper appreciation for the elegance and interconnectedness of mathematical principles. The journey through logarithmic equations highlights the importance of understanding the basic rules and applying them systematically to solve problems. Whether it's using the quotient rule to simplify differences of logarithms, the product rule to combine sums of logarithms, or the power rule to handle exponents within logarithms, each rule plays a crucial role in the manipulation and simplification process. The ability to recognize when and how to apply these rules is a key skill for anyone studying mathematics. Furthermore, the examples discussed in this article illustrate that logarithms are not just abstract mathematical concepts but powerful tools for solving real-world problems. Logarithmic scales, for instance, are used in various fields such as acoustics, seismology, and chemistry to represent and analyze data that span several orders of magnitude. The principles we’ve covered here lay the groundwork for more advanced mathematical concepts and applications. As you continue your mathematical journey, the understanding of logarithms and their properties will undoubtedly prove invaluable. This article aims to provide not just the solutions to specific problems but a comprehensive understanding of the underlying principles, empowering you to approach new challenges with confidence and skill. By mastering these fundamentals, you are well-equipped to explore the vast and fascinating world of mathematics further.