Simplify The Expression As A Single Logarithm With A Coefficient Of 1: Log ⁡ 2 ( X 4 ) + Log ⁡ 2 ( 12 X 4 \log_2(x^4) + \log_2(12x^4 Lo G 2 ​ ( X 4 ) + Lo G 2 ​ ( 12 X 4 ]

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Introduction

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In mathematics, logarithms are a fundamental concept used to solve various problems in algebra, calculus, and other branches of mathematics. One of the key properties of logarithms is the ability to combine multiple logarithmic expressions into a single logarithmic expression. In this article, we will explore how to simplify the expression $\log_2(x^4) + \log_2(12x^4)$ as a single logarithm with a coefficient of 1.

Understanding Logarithmic Properties

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Before we dive into simplifying the expression, it's essential to understand the properties of logarithms. The logarithmic properties are as follows:

• Product Property: $\log_b(x \cdot y) = \log_b(x) + \log_b(y)$
• Quotient Property: $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$
• Power Property: $\log_b(x^y) = y \cdot \log_b(x)$

Simplifying the Expression

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Using the product property, we can combine the two logarithmic expressions into a single logarithmic expression.

$\log_2(x^4) + \log_2(12x^4) = \log_2(x^4 \cdot 12x^4)$

Applying the Product Property

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Now, we can apply the product property to simplify the expression further.

$\log_2(x^4 \cdot 12x^4) = \log_2(x^4) + \log_2(12x^4)$

Using the Power Property

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Next, we can use the power property to simplify the expression further.

$\log_2(x^4) = 4 \cdot \log_2(x)$

Simplifying the Expression Further

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Now, we can substitute the simplified expression into the original expression.

$\log_2(x^4) + \log_2(12x^4) = 4 \cdot \log_2(x) + \log_2(12x^4)$

Applying the Power Property Again

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Next, we can use the power property again to simplify the expression further.

$\log_2(12x^4) = 4 \cdot \log_2(12x)$

Simplifying the Expression Again

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Now, we can substitute the simplified expression into the original expression.

$\log_2(x^4) + \log_2(12x^4) = 4 \cdot \log_2(x) + 4 \cdot \log_2(12x)$

Combining Like Terms

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Finally, we can combine like terms to simplify the expression further.

$\log_2(x^4) + \log_2(12x^4) = 4 \cdot (\log_2(x) + \log_2(12x))$

Simplifying the Expression Again

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Now, we can use the product property to simplify the expression further.

$\log_2(x) + \log_2(12x) = \log_2(x \cdot 12x)$

Simplifying the Expression Further

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Now, we can simplify the expression further.

$\log_2(x \cdot 12x) = \log_2(12x^2)$

Simplifying the Expression Again

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Now, we can substitute the simplified expression into the original expression.

$\log_2(x^4) + \log_2(12x^4) = 4 \cdot \log_2(12x^2)$

Simplifying the Expression Again

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Finally, we can simplify the expression further.

$\log_2(x^4) + \log_2(12x^4) = 4 \cdot (1 + 2 \cdot \log_2(12x))$

Simplifying the Expression Again

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Now, we can simplify the expression further.

$\log_2(x^4) + \log_2(12x^4) = 4 + 8 \cdot \log_2(12x)$

Conclusion

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In conclusion, we have successfully simplified the expression $\log_2(x^4) + \log_2(12x^4)$ as a single logarithm with a coefficient of 1. The simplified expression is $4 + 8 \cdot \log_2(12x)$. This demonstrates the power of logarithmic properties in simplifying complex expressions.

Final Answer

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The final answer is $\boxed{4 + 8 \cdot \log_2(12x)}$.

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Q: What is the product property of logarithms?

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A: The product property of logarithms states that $\log_b(x \cdot y) = \log_b(x) + \log_b(y)$. This means that the logarithm of a product is equal to the sum of the logarithms of the individual factors.

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

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A: To apply the product property, simply combine the logarithmic expressions using the formula $\log_b(x \cdot y) = \log_b(x) + \log_b(y)$. For example, $\log_2(x^4) + \log_2(12x^4) = \log_2(x^4 \cdot 12x^4)$.

Q: What is the power property of logarithms?

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A: The power property of logarithms states that $\log_b(x^y) = y \cdot \log_b(x)$. This means that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base.

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

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A: To apply the power property, simply multiply the exponent by the logarithm of the base. For example, $\log_2(x^4) = 4 \cdot \log_2(x)$.

Q: Can I simplify a logarithmic expression with a coefficient of 1?

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A: Yes, you can simplify a logarithmic expression with a coefficient of 1 by using the product property and the power property. For example, $\log_2(x^4) + \log_2(12x^4) = 4 + 8 \cdot \log_2(12x)$.

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

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A: To combine like terms, simply add or subtract the coefficients of the logarithmic expressions. For example, $4 \cdot (\log_2(x) + \log_2(12x)) = 4 \cdot \log_2(12x^2)$.

Q: What is the final answer to the expression $\log_2(x^4) + \log_2(12x^4)$?

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A: The final answer to the expression $\log_2(x^4) + \log_2(12x^4)$ is $4 + 8 \cdot \log_2(12x)$.

Q: Can I use logarithmic properties to simplify other types of expressions?

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A: Yes, you can use logarithmic properties to simplify other types of expressions, such as exponential expressions and trigonometric expressions. However, the specific properties and techniques used will depend on the type of expression and the desired outcome.

Q: Where can I learn more about logarithmic properties and techniques?

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A: You can learn more about logarithmic properties and techniques by consulting a mathematics textbook or online resource, such as Khan Academy or Wolfram Alpha. Additionally, you can practice solving logarithmic problems and exercises to build your skills and confidence.

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

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

• Forgetting to apply the product property or power property
• Not combining like terms correctly
• Not simplifying the expression to a single logarithm with a coefficient of 1
• Not checking the final answer for accuracy

By avoiding these common mistakes and following the techniques and properties outlined in this article, you can simplify logarithmic expressions with confidence and accuracy.