Fill In The Missing Values In Logarithmic Equations A Step By Step Guide

In the realm of mathematics, logarithmic equations hold a significant position, especially when dealing with exponential relationships. These equations are essentially the inverse of exponential functions and provide a powerful tool for solving problems involving exponents. Before diving into filling in the missing values, it's crucial to grasp the fundamental principles and properties of logarithms. A logarithm answers the question: "To what power must we raise a certain base to obtain a specific number?" For instance, if we have the equation $\log_b a = c$, it means that $b$ raised to the power of $c$ equals $a$ (i.e., $b^c = a$). The base, $b$, is the foundation upon which the logarithm is built, and it plays a critical role in determining the value of the logarithm. Understanding the base and its relationship to the argument (the number we're taking the logarithm of) is key to manipulating and solving logarithmic equations. The argument, in this case, is $a$, and it must be a positive number. Logarithms are undefined for non-positive numbers because there's no power to which we can raise a positive base to get a zero or a negative result. Furthermore, the result, $c$, represents the exponent or the power to which we must raise the base to get the argument. This exponent is the very essence of what a logarithm reveals. Grasping the connection between the base, argument, and exponent is paramount for solving logarithmic equations efficiently and accurately. One of the most crucial aspects of mastering logarithms is understanding their properties. These properties act as powerful tools that allow us to simplify complex expressions, combine multiple logarithmic terms, and ultimately solve equations that might otherwise seem insurmountable. The primary properties of logarithms include the product rule, the quotient rule, and the power rule. These rules are derived directly from the properties of exponents, reflecting the inverse relationship between logarithms and exponential functions. 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 $\log_b (mn) = \log_b m + \log_b n$. This rule is invaluable when we encounter the logarithm of a product within an equation. The quotient rule, on the other hand, deals with the logarithm of a quotient. It states that the logarithm of a quotient is equal to the difference between the logarithm of the numerator and the logarithm of the denominator, expressed as $\log_b (m/n) = \log_b m - \log_b n$. This rule is particularly useful when simplifying expressions involving fractions within logarithms. Lastly, the power rule addresses the logarithm of a number raised to a power. It states that the logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number, represented as $\log_b (m^p) = p \log_b m$. This rule is essential for dealing with exponents within logarithmic expressions. By effectively applying these properties, we can transform complicated logarithmic equations into simpler, more manageable forms, paving the way for accurate solutions.

(a) $\log _2 5+\log _2 11=\log _2 \square$

To solve this logarithmic equation, we will leverage the properties of logarithms, specifically the product rule. The product rule states that the sum of the logarithms of two numbers with the same base is equal to the logarithm of the product of those numbers. In mathematical terms, this is expressed as: $\log_b m + \log_b n = \log_b (mn)$. This rule is a cornerstone of logarithmic manipulations, allowing us to combine logarithmic terms and simplify equations. In our specific problem, we have the equation: $\log _2 5+\log _2 11=\log _2 \square$. We can see that we have the sum of two logarithms with the same base, which is 2. The arguments of the logarithms are 5 and 11. Applying the product rule, we can combine these two logarithmic terms into a single logarithm. We multiply the arguments, 5 and 11, together: $5 \times 11 = 55$. This means that the right-hand side of the equation becomes the logarithm of 55 to the base 2. So, we have: $\log _2 55$. Now we can rewrite the original equation as: $\log _2 5+\log _2 11 = \log _2 55$. Comparing this with the original equation, $\log _2 5+\log _2 11=\log _2 \square$, we can clearly see that the missing value, represented by the square, is 55. Therefore, the complete equation is: $\log _2 5+\log _2 11=\log _2 55$. This solution demonstrates the power and elegance of the product rule in simplifying logarithmic expressions. By understanding and applying this rule, we can effectively handle equations involving sums of logarithms, transforming them into more manageable forms. The product rule, along with the quotient and power rules, forms the foundation of logarithmic manipulations, enabling us to solve a wide range of logarithmic problems. In this case, the application of the product rule directly led us to the solution, filling in the missing value and completing the equation. The ability to recognize and apply these logarithmic properties is crucial for success in solving logarithmic equations and understanding the behavior of logarithmic functions. By mastering these fundamental concepts, students can approach more complex problems with confidence and achieve a deeper understanding of the mathematical relationships underlying logarithms.

(b) $\log _5 9-\log _5 \square=\log _5 \frac{9}{11}$

This part of the problem requires us to apply another key property of logarithms: the quotient rule. The quotient rule states that the logarithm of a quotient is equal to the difference between the logarithm of the numerator and the logarithm of the denominator, provided that all logarithms have the same base. Mathematically, this is expressed as: $\log_b m - \log_b n = \log_b (\frac{m}{n})$. This rule is particularly useful when dealing with equations that involve differences of logarithms. It allows us to combine these differences into a single logarithm of a quotient, simplifying the equation and making it easier to solve. The equation we are given is: $\log _5 9-\log _5 \square=\log _5 \frac{9}{11}$. We can see that we have the difference of two logarithms with the same base, which is 5. The first term has an argument of 9, and we need to find the missing argument in the second term. The right-hand side of the equation is $\log _5 \frac{9}{11}$, which is the logarithm of a quotient. Applying the quotient rule in reverse, we can think of the left-hand side as the logarithm of a quotient. Let's denote the missing value in the square as $x$. Then the equation can be rewritten as: $\log _5 9 - \log _5 x = \log _5 (\frac{9}{x})$. Now we can equate this to the right-hand side of the original equation: $\log _5 (\frac{9}{x}) = \log _5 \frac{9}{11}$. Since the logarithms have the same base, we can equate the arguments: $\frac{9}{x} = \frac{9}{11}$. To solve for $x$, we can cross-multiply or simply observe that if the numerators are the same, the denominators must also be the same for the fractions to be equal. Therefore, $x = 11$. This means that the missing value in the square is 11. So the complete equation is: $\log _5 9-\log _5 11=\log _5 \frac{9}{11}$. This solution highlights the importance of the quotient rule in simplifying logarithmic expressions. By applying the quotient rule, we were able to transform the difference of two logarithms into the logarithm of a quotient, which then allowed us to solve for the missing value. The ability to apply the quotient rule, as well as the product and power rules, is essential for effectively manipulating logarithmic equations and finding solutions. This problem demonstrates how a solid understanding of logarithmic properties can lead to a straightforward and elegant solution. Students who master these properties will be well-equipped to tackle a wide range of logarithmic problems and applications. The quotient rule, in particular, is a powerful tool for simplifying expressions involving fractions and differences of logarithms, making it an indispensable part of the mathematician's toolkit.

(c) $\log _9 25=\square \log _9 5$

In this problem, we encounter yet another fundamental property of logarithms: the power rule. The power rule is a powerful tool that allows us to deal with exponents within logarithmic expressions. It states that the logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number. In mathematical terms, this is expressed as: $\log_b (m^p) = p \log_b m$. This rule is particularly useful when we need to simplify expressions or solve equations involving logarithms of numbers raised to powers. The given equation is: $\log _9 25=\square \log _9 5$. We need to find the missing value that, when multiplied by $\log _9 5$, will give us $\log _9 25$. To solve this, we can rewrite 25 as a power of 5. We know that $25 = 5^2$. Substituting this into the equation, we get: $\log _9 (5^2) = \square \log _9 5$. Now we can apply the power rule to the left-hand side of the equation. The power rule tells us that the logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number. In this case, we have $\log _9 (5^2)$, which, according to the power rule, is equal to $2 \log _9 5$. So, we can rewrite the equation as: $2 \log _9 5 = \square \log _9 5$. Now we can see that the left-hand side and the right-hand side have a common factor of $\log _9 5$. To find the missing value, we can simply compare the coefficients of $\log _9 5$ on both sides of the equation. The coefficient on the left-hand side is 2, and the coefficient on the right-hand side is the missing value represented by the square. Therefore, the missing value is 2. So the complete equation is: $\log _9 25=2 \log _9 5$. This solution elegantly demonstrates the application of the power rule in simplifying logarithmic equations. By recognizing that 25 can be expressed as a power of 5 and then applying the power rule, we were able to easily find the missing value. The power rule is an essential tool in logarithmic manipulations, allowing us to simplify expressions and solve equations that might otherwise seem complex. The ability to apply the power rule, as well as the product and quotient rules, is crucial for effectively working with logarithms. This problem highlights how a solid understanding of logarithmic properties can lead to a straightforward solution. Students who master these properties will be well-prepared to tackle a wide range of logarithmic problems and applications. The power rule, in particular, is a versatile tool for simplifying expressions involving exponents within logarithms, making it an indispensable part of the mathematician's toolkit. By understanding and applying these fundamental concepts, students can approach more complex problems with confidence and achieve a deeper understanding of the mathematical relationships underlying logarithms.

By mastering these logarithmic properties – the product rule, the quotient rule, and the power rule – and understanding their applications, you can confidently solve a wide variety of logarithmic equations and problems. These rules act as powerful tools for simplifying complex expressions and manipulating logarithmic terms, making them essential for anyone working with logarithmic functions. Remember to always pay attention to the base of the logarithm and apply the rules carefully to ensure accurate results. With practice and a solid grasp of these principles, you'll be well-equipped to tackle any logarithmic challenge.