Solving Logarithmic Expressions A Step By Step Guide

In the realm of mathematics, logarithmic expressions often present a unique challenge, requiring a blend of understanding logarithmic properties and careful application of these principles. One such expression is \log rac{14}{3}+\log rac{11}{5}-\log rac{22}{15}. This problem appears deceptively simple, yet it necessitates a systematic approach to unravel its intricacies. This comprehensive guide will walk you through the step-by-step process of solving this logarithmic equation, providing clarity and insights into the underlying concepts.

Understanding Logarithmic Properties

Before diving into the solution, it's crucial to grasp the fundamental logarithmic properties that govern these expressions. Logarithms, at their core, are the inverse operations 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. For instance, $\log_{10} 100 = 2$ because 10 raised to the power of 2 equals 100. Understanding this basic definition is paramount to manipulating logarithmic expressions effectively.

The logarithmic properties that are most relevant to this problem are the product rule, the quotient rule, and the power 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: $\log_b (mn) = \log_b m + \log_b n$. Conversely, the quotient rule states that the logarithm of a quotient is equal to the difference between the logarithms of the numerator and the denominator: $\log_b (m/n) = \log_b m - \log_b n$. These two rules are essential tools in simplifying and combining logarithmic terms. The power rule, while not directly used in this specific problem, is still a key concept: $\log_b (m^p) = p \log_b m$. This rule allows us to deal with exponents within logarithms.

Furthermore, it's important to remember that when the base of the logarithm is not explicitly stated, it is generally assumed to be base 10. This is known as the common logarithm and is often written as just “log” without the subscript. With these properties in mind, we are well-equipped to tackle the given expression.

Step-by-Step Solution

Now, let's dissect the expression \log rac{14}{3}+\log rac{11}{5}-\log rac{22}{15} step by step. Our goal is to simplify this expression into a single logarithmic term.

Step 1: Applying the Product Rule

The first step involves applying the product rule to the first two terms of the expression. We have \log rac{14}{3}+\log rac{11}{5}. According to the product rule, we can combine these two logarithms into a single logarithm by multiplying their arguments. This gives us:

\log rac{14}{3}+\log rac{11}{5} = \log ig(rac{14}{3} imes rac{11}{5}ig)

Now, we perform the multiplication within the logarithm:

\log ig(rac{14}{3} imes rac{11}{5}ig) = \log rac{14 imes 11}{3 imes 5} = \log rac{154}{15}

So, the expression now becomes: \log rac{154}{15} - \log rac{22}{15}. This simplification has reduced the three logarithmic terms into two, making the expression more manageable.

Step 2: Applying the Quotient Rule

Next, we apply the quotient rule to combine the two remaining logarithmic terms. The quotient rule states that the difference between two logarithms is equal to the logarithm of the quotient of their arguments. In our case, we have \log rac{154}{15} - \log rac{22}{15}. Applying the quotient rule, we get:

\log rac{154}{15} - \log rac{22}{15} = \log ig(rac{154/15}{22/15}ig)

Now, we need to simplify the fraction inside the logarithm. Dividing by a fraction is the same as multiplying by its reciprocal, so we have:

\log ig(rac{154/15}{22/15}ig) = \log ig(rac{154}{15} imes rac{15}{22}ig)

Step 3: Simplifying the Fraction

Now, we can simplify the fraction inside the logarithm by canceling out common factors. We have:

\log ig(rac{154}{15} imes rac{15}{22}ig) = \log ig(rac{154 imes 15}{15 imes 22}ig)

The 15 in the numerator and denominator cancel each other out:

\log ig(rac{154 imes 15}{15 imes 22}ig) = \log rac{154}{22}

Now, we simplify the fraction $\frac{154}{22}$. Both 154 and 22 are divisible by 22. Dividing both the numerator and the denominator by 22, we get:

$\frac{154}{22} = \frac{154 ext{ ÷ } 22}{22 ext{ ÷ } 22} = \frac{7}{1} = 7$

So, the expression simplifies to:

\log rac{154}{22} = \log 7

Final Answer

Therefore, the simplified form of the expression \log rac{14}{3}+\log rac{11}{5}-\log rac{22}{15} is $\log 7$. This means that the correct answer to fill in the box is 7.

Common Mistakes to Avoid

When working with logarithmic expressions, several common mistakes can lead to incorrect answers. Being aware of these pitfalls can help you approach problems more confidently and accurately.

One frequent mistake is misapplying the logarithmic properties. For example, students might incorrectly try to apply the product rule to terms that are being subtracted or the quotient rule to terms that are being added. It is crucial to remember that the product rule applies only to the sum of logarithms, and the quotient rule applies only to the difference of logarithms. Similarly, the power rule should only be used when an exponent is present within the logarithm.

Another common error is mishandling fractions within logarithms. When applying the quotient rule, it’s essential to correctly divide the arguments of the logarithms. This often involves multiplying by the reciprocal of a fraction, and any errors in this step can propagate through the rest of the solution. Always double-check your fraction manipulations to ensure accuracy.

Simplifying fractions prematurely can also lead to mistakes. It’s often best to combine logarithmic terms first and then simplify the resulting fraction. This ensures that all factors are properly accounted for and reduces the risk of overlooking common factors.

Finally, a lack of understanding of the fundamental definition of logarithms can be a significant obstacle. Remember that a logarithm is an exponent, and understanding this relationship is key to grasping logarithmic properties and solving logarithmic equations. If you find yourself struggling with logarithmic problems, revisiting the basic definition and properties can be incredibly helpful.

Practice Problems

To solidify your understanding of logarithmic expressions, working through practice problems is invaluable. Here are a few additional problems that you can try:

1. Simplify: $\log 2 + \log 3 - \log 5$
2. Simplify: $2 \log x + \log y - \log z$
3. Simplify: \log rac{25}{4} + \log rac{16}{5} - \log 20

By tackling these problems, you’ll gain confidence in applying logarithmic properties and become more adept at simplifying complex expressions. Remember to approach each problem systematically, breaking it down into manageable steps and carefully applying the relevant logarithmic rules.

Conclusion

Mastering logarithmic expressions requires a solid understanding of logarithmic properties and a methodical approach to problem-solving. By applying the product rule, the quotient rule, and other key concepts, you can simplify complex expressions and arrive at accurate solutions. The expression \log rac{14}{3}+\log rac{11}{5}-\log rac{22}{15} serves as an excellent example of how these principles can be applied to solve logarithmic problems. Remember to avoid common mistakes and to practice regularly to enhance your skills. With perseverance and a clear understanding of the fundamentals, you can confidently navigate the world of logarithms.

In summary, the solution to \log rac{14}{3}+\log rac{11}{5}-\log rac{22}{15} is $\log 7$, and the correct answer to fill in the box is 7. Keep practicing, and you'll become proficient in simplifying logarithmic expressions!