Type The Correct Answer In The Box. Log ⁡ 14 3 + Log ⁡ 11 5 − Log ⁡ 22 15 = Log ⁡ \log \frac{14}{3} + \log \frac{11}{5} - \log \frac{22}{15} = \log Lo G 3 14 ​ + Lo G 5 11 ​ − Lo G 15 22 ​ = Lo G

Introduction

Logarithmic expressions can be complex and challenging to simplify, but with the right techniques and understanding, they can be broken down into manageable parts. In this article, we will explore how to simplify a logarithmic expression involving multiple terms, using the properties of logarithms.

Understanding Logarithmic Properties

Before we dive into the problem, 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 crucial in simplifying the given expression.

The Problem

The problem we are given is:

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

Our goal is to simplify the left-hand side of the equation and find the value of the logarithm.

Step 1: Apply the Quotient Property

The first step is to apply the quotient property to each term in the expression. This will help us simplify the fractions inside the logarithms.

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

Now, we can rewrite the original expression using these simplified fractions.

Step 2: Combine Like Terms

The next step is to combine like terms in the expression. We can do this by adding or subtracting the logarithms of the fractions.

$\log 14 - \log 3 + \log 11 - \log 5 - \log 22 + \log 15$

Now, we can group the like terms together.

$(\log 14 + \log 11 - \log 22) + (\log 3 - \log 5 + \log 15)$

Step 3: Apply the Product Property

The product property states that the logarithm of a product is equal to the sum of the logarithms. We can apply this property to the first group of terms.

$\log 14 + \log 11 - \log 22 = \log (14 \cdot 11) - \log 22 = \log 154 - \log 22$

Similarly, we can apply the product property to the second group of terms.

$\log 3 - \log 5 + \log 15 = \log 3 - \log 5 + \log (5 \cdot 3) = \log 3 - \log 5 + \log 15$

Now, we can rewrite the expression using these simplified terms.

Step 4: Simplify the Expression

The final step is to simplify the expression by combining like terms.

$\log 154 - \log 22 + \log 3 - \log 5 + \log 15$

We can simplify this expression by combining the logarithms of the fractions.

$\log \frac{154}{22} + \log \frac{3}{5} + \log 15$

Now, we can apply the quotient property to simplify the fractions.

$\log \frac{154}{22} = \log 7$ $\log \frac{3}{5} = \log 3 - \log 5$

Substituting these simplified fractions back into the expression, we get:

$\log 7 + \log 3 - \log 5 + \log 15$

Step 5: Apply the Product Property Again

The product property states that the logarithm of a product is equal to the sum of the logarithms. We can apply this property to the expression.

$\log 7 + \log 3 - \log 5 + \log 15 = \log (7 \cdot 3 \cdot 15) - \log 5$

Now, we can simplify the expression by combining the logarithms of the fractions.

$\log (7 \cdot 3 \cdot 15) - \log 5 = \log 315 - \log 5$

Step 6: Simplify the Final Expression

The final step is to simplify the expression by combining like terms.

$\log 315 - \log 5 = \log \frac{315}{5}$

Now, we can simplify the fraction inside the logarithm.

$\log \frac{315}{5} = \log 63$

Conclusion

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Q&A: Simplifying Logarithmic Expressions

Q: What are the 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 need to rewrite the fraction inside the logarithm as a difference of logarithms. For example:

$\log \frac{14}{3} = \log 14 - \log 3$

Q: How do I combine like terms in a logarithmic expression?

A: To combine like terms, you need to group the terms with the same logarithm together and add or subtract them. For example:

$\log 14 - \log 3 + \log 11 - \log 5 - \log 22 + \log 15$

Can be rewritten as:

$(\log 14 + \log 11 - \log 22) + (\log 3 - \log 5 + \log 15)$

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

A: To apply the product property, you need to rewrite the product inside the logarithm as a sum of logarithms. For example:

$\log (14 \cdot 11) = \log 14 + \log 11$

Q: What is the final simplified expression for the given problem?

A: The final simplified expression is $\log 63$.

Q: How do I check my work when simplifying a logarithmic expression?

A: To check your work, you can plug in the original expression and the simplified expression into a calculator or use a logarithmic table to verify that they are equal.

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

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

• Not applying the quotient property correctly
• Not combining like terms correctly
• Not applying the product property correctly
• Not checking your work

Q: How do I apply the properties of logarithms to simplify more complex logarithmic expressions?

A: To apply the properties of logarithms to simplify more complex logarithmic expressions, you need to:

• Identify the properties that can be applied to the expression
• Apply the properties in the correct order
• Simplify the expression using the properties
• Check your work to ensure that the expression is simplified correctly

Conclusion

In this article, we provided a step-by-step guide to simplifying logarithmic expressions using the properties of logarithms. We also answered some common questions about simplifying logarithmic expressions and provided tips for avoiding common mistakes. By following these steps and tips, you can simplify even the most complex logarithmic expressions with ease.