Type The Correct Answer In The Box.${ \log \frac{14}{3} + \log \frac{11}{5} - \log \frac{22}{15} = \log }$

Introduction

Logarithmic expressions can be complex and challenging to simplify. However, with a clear understanding of the properties of logarithms, we can break down these expressions into manageable parts. In this article, we will explore how to simplify a given logarithmic expression using the properties of logarithms.

Understanding Logarithmic Properties

Before we dive into the solution, it's essential to understand the properties of logarithms. The three main properties of logarithms are:

• Product Property: $\log (ab) = \log a + \log b$
• Quotient Property: $\log \frac{a}{b} = \log a - \log b$
• Power Property: $\log a^b = b \log a$

These properties will be the foundation of our solution.

Simplifying the Logarithmic Expression

The given logarithmic expression is:

$\log \frac{14}{3} + \log \frac{11}{5} - \log \frac{22}{15} = \log x$

Our goal is to simplify this expression and find the value of $x$.

Step 1: Apply the Quotient Property

We can start by applying the quotient property to the first two terms:

$\log \frac{14}{3} + \log \frac{11}{5} = \log \left(\frac{14}{3} \cdot \frac{11}{5}\right)$

Using the product property, we can simplify the expression inside the logarithm:

$\log \left(\frac{14}{3} \cdot \frac{11}{5}\right) = \log \left(\frac{154}{15}\right)$

Step 2: Apply the Quotient Property Again

Now, we can apply the quotient property to the simplified expression and the third term:

$\log \left(\frac{154}{15}\right) - \log \frac{22}{15} = \log \left(\frac{154}{15} \div \frac{22}{15}\right)$

Using the quotient property, we can simplify the expression inside the logarithm:

$\log \left(\frac{154}{15} \div \frac{22}{15}\right) = \log \left(\frac{154}{15} \cdot \frac{15}{22}\right)$

Step 3: Simplify the Expression

Now, we can simplify the expression inside the logarithm:

$\log \left(\frac{154}{15} \cdot \frac{15}{22}\right) = \log \left(\frac{154}{22}\right)$

Step 4: Simplify the Fraction

We can simplify the fraction inside the logarithm:

$\log \left(\frac{154}{22}\right) = \log \left(7\right)$

Conclusion

In this article, we simplified a given logarithmic expression using the properties of logarithms. We applied the quotient property to break down the expression into manageable parts and then simplified the resulting expression. The final answer is $\boxed{7}$.

Additional Examples

Here are a few additional examples of simplifying logarithmic expressions using the properties of logarithms:

• $\log \frac{24}{8} + \log \frac{9}{4} - \log \frac{12}{5} = \log x$
• $\log \frac{36}{12} + \log \frac{25}{10} - \log \frac{15}{8} = \log x$
• $\log \frac{48}{16} + \log \frac{49}{7} - \log \frac{21}{14} = \log x$

These examples demonstrate how to apply the properties of logarithms to simplify complex expressions.

Final Thoughts

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Q: What are the three main properties of logarithms?

A: The three main properties of logarithms are:

• Product Property: $\log (ab) = \log a + \log b$
• Quotient Property: $\log \frac{a}{b} = \log a - \log b$
• Power Property: $\log a^b = b \log a$

Q: How do I apply the quotient property to simplify a logarithmic expression?

A: To apply the quotient property, you can use the following steps:

1. Identify the quotient property in the expression.
2. Rewrite the expression using the quotient property.
3. Simplify the resulting expression.

For example, if you have the expression $\log \frac{14}{3} + \log \frac{11}{5} - \log \frac{22}{15}$, you can apply the quotient property as follows:

$\log \frac{14}{3} + \log \frac{11}{5} - \log \frac{22}{15} = \log \left(\frac{14}{3} \cdot \frac{11}{5}\right) - \log \frac{22}{15}$

Q: How do I simplify a logarithmic expression with multiple terms?

A: To simplify a logarithmic expression with multiple terms, you can use the following steps:

1. Identify the terms in the expression.
2. Apply the quotient property to combine the terms.
3. Simplify the resulting expression.

For example, if you have the expression $\log \frac{24}{8} + \log \frac{9}{4} - \log \frac{12}{5}$, you can simplify it as follows:

$\log \frac{24}{8} + \log \frac{9}{4} - \log \frac{12}{5} = \log \left(\frac{24}{8} \cdot \frac{9}{4}\right) - \log \frac{12}{5}$

Q: What is the difference between the product property and the quotient property?

A: The product property states that $\log (ab) = \log a + \log b$, while the quotient property states that $\log \frac{a}{b} = \log a - \log b$. The product property is used to combine two or more terms inside a logarithm, while the quotient property is used to divide two or more terms inside a logarithm.

Q: Can I use the power property to simplify a logarithmic expression?

A: Yes, you can use the power property to simplify a logarithmic expression. The power property states that $\log a^b = b \log a$. You can use this property to simplify expressions with exponents inside the logarithm.

For example, if you have the expression $\log 2^3$, you can simplify it as follows:

$\log 2^3 = 3 \log 2$

Q: What are some common mistakes to avoid when simplifying logarithmic expressions?

A: Some common mistakes to avoid when simplifying logarithmic expressions include:

• Forgetting to apply the quotient property when dividing terms inside a logarithm.
• Forgetting to apply the product property when combining terms inside a logarithm.
• Not simplifying the expression inside the logarithm.
• Not using the correct property of logarithms to simplify the expression.

By avoiding these common mistakes, you can ensure that your logarithmic expressions are simplified correctly.

Conclusion

Simplifying logarithmic expressions can be challenging, but with a clear understanding of the properties of logarithms, you can break down these expressions into manageable parts. By applying the quotient property, product property, and power property, you can simplify complex expressions and find the value of $x$. Remember to avoid common mistakes and use the correct properties of logarithms to simplify your expressions.