Simplify The Expression:$\[ 1 + 2 \log N + 3 \log N + 3 \log N \times 4 \\]

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Introduction

In mathematics, simplifying expressions is a crucial step in solving problems and understanding complex concepts. The given expression, $1 + 2 \log n + 3 \log n + 3 \log n \times 4$, appears to be a combination of logarithmic and arithmetic operations. In this article, we will simplify the expression step by step, using the properties of logarithms and basic algebra.

Understanding the Expression

The given expression consists of four terms:

1. $1$
2. $2 \log n$
3. $3 \log n$
4. $3 \log n \times 4$

The first term is a constant, while the remaining three terms involve logarithmic functions. The logarithmic functions are in base 10, as indicated by the absence of any other base notation.

Simplifying the Expression

To simplify the expression, we will start by combining the logarithmic terms.

Combining Logarithmic Terms

Using the property of logarithms that states $\log a + \log b = \log (a \times b)$, we can combine the second, third, and fourth terms as follows:

$$2 \log n + 3 \log n + 3 \log n \times 4 = 2 \log n + 3 \log n + 12 \log n$$

Now, we can combine the logarithmic terms using the property mentioned above:

$$2 \log n + 3 \log n + 12 \log n = (2 + 3 + 12) \log n$$

Simplifying the coefficient of the logarithmic term, we get:

$$(2 + 3 + 12) \log n = 17 \log n$$

Simplifying the Expression Further

Now that we have combined the logarithmic terms, we can simplify the expression further by adding the constant term to the combined logarithmic term:

$$1 + 17 \log n$$

This is the simplified expression.

Conclusion

In this article, we simplified the given expression, $1 + 2 \log n + 3 \log n + 3 \log n \times 4$, using the properties of logarithms and basic algebra. We combined the logarithmic terms and simplified the expression further by adding the constant term to the combined logarithmic term. The simplified expression is $1 + 17 \log n$.

Final Answer

The final answer is $\boxed{1 + 17 \log n}$.

Step-by-Step Solution

Here is the step-by-step solution to the problem:

1. Combine the logarithmic terms using the property of logarithms: $2 \log n + 3 \log n + 3 \log n \times 4 = 2 \log n + 3 \log n + 12 \log n$
2. Combine the logarithmic terms: $2 \log n + 3 \log n + 12 \log n = (2 + 3 + 12) \log n$
3. Simplify the coefficient of the logarithmic term: $(2 + 3 + 12) \log n = 17 \log n$
4. Add the constant term to the combined logarithmic term: $1 + 17 \log n$

Frequently Asked Questions

• What is the simplified expression?
• How do you combine logarithmic terms?
• What is the property of logarithms used to combine logarithmic terms?

Answer to Frequently Asked Questions

• The simplified expression is $1 + 17 \log n$.
• To combine logarithmic terms, you use the property of logarithms that states $\log a + \log b = \log (a \times b)$.
• The property of logarithms used to combine logarithmic terms is $\log a + \log b = \log (a \times b)$.

Related Topics

• Properties of logarithms
• Simplifying expressions
• Algebraic manipulations

References

• [1] "Logarithms" by Khan Academy
• [2] "Simplifying Expressions" by Mathway
• [3] "Algebraic Manipulations" by Wolfram MathWorld
  

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Introduction

In our previous article, we simplified the expression $1 + 2 \log n + 3 \log n + 3 \log n \times 4$ using the properties of logarithms and basic algebra. In this article, we will answer some frequently asked questions related to the simplification of the expression.

Q&A

Q: What is the simplified expression?

A: The simplified expression is $1 + 17 \log n$.

Q: How do you combine logarithmic terms?

A: To combine logarithmic terms, you use the property of logarithms that states $\log a + \log b = \log (a \times b)$. This property allows you to combine multiple logarithmic terms into a single logarithmic term.

Q: What is the property of logarithms used to combine logarithmic terms?

A: The property of logarithms used to combine logarithmic terms is $\log a + \log b = \log (a \times b)$. This property is a fundamental concept in logarithmic algebra and is used extensively in mathematics and engineering.

Q: How do you simplify an expression with multiple logarithmic terms?

A: To simplify an expression with multiple logarithmic terms, you can use the property of logarithms mentioned above to combine the logarithmic terms. Once you have combined the logarithmic terms, you can simplify the expression further by adding or subtracting the constant terms.

Q: What is the difference between a logarithmic term and a constant term?

A: A logarithmic term is an expression that involves a logarithm, such as $\log n$ or $2 \log n$. A constant term is an expression that does not involve a logarithm, such as $1$ or $3$.

Q: How do you add or subtract logarithmic terms?

A: To add or subtract logarithmic terms, you can use the property of logarithms that states $\log a + \log b = \log (a \times b)$ or $\log a - \log b = \log \left(\frac{a}{b}\right)$. This property allows you to combine multiple logarithmic terms into a single logarithmic term.

Q: What is the importance of simplifying expressions?

A: Simplifying expressions is an important step in solving problems and understanding complex concepts. By simplifying expressions, you can make them easier to work with and understand, which can help you to solve problems more efficiently and effectively.

Conclusion

In this article, we answered some frequently asked questions related to the simplification of the expression $1 + 2 \log n + 3 \log n + 3 \log n \times 4$. We discussed the property of logarithms used to combine logarithmic terms, how to simplify an expression with multiple logarithmic terms, and the importance of simplifying expressions.

Final Answer

The final answer is $\boxed{1 + 17 \log n}$.

Step-by-Step Solution

Here is the step-by-step solution to the problem:

1. Combine the logarithmic terms using the property of logarithms: $2 \log n + 3 \log n + 3 \log n \times 4 = 2 \log n + 3 \log n + 12 \log n$
2. Combine the logarithmic terms: $2 \log n + 3 \log n + 12 \log n = (2 + 3 + 12) \log n$
3. Simplify the coefficient of the logarithmic term: $(2 + 3 + 12) \log n = 17 \log n$
4. Add the constant term to the combined logarithmic term: $1 + 17 \log n$

Frequently Asked Questions

• What is the simplified expression?
• How do you combine logarithmic terms?
• What is the property of logarithms used to combine logarithmic terms?

Answer to Frequently Asked Questions

• The simplified expression is $1 + 17 \log n$.
• To combine logarithmic terms, you use the property of logarithms that states $\log a + \log b = \log (a \times b)$.
• The property of logarithms used to combine logarithmic terms is $\log a + \log b = \log (a \times b)$.

Related Topics

• Properties of logarithms
• Simplifying expressions
• Algebraic manipulations

References

• [1] "Logarithms" by Khan Academy
• [2] "Simplifying Expressions" by Mathway
• [3] "Algebraic Manipulations" by Wolfram MathWorld