Evaluate The Expression: \log _2\left(32^2\right ]

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Introduction

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In this article, we will delve into the world of logarithms and evaluate the expression $\log _2\left(32^2\right)$. Logarithms are a fundamental concept in mathematics, and understanding how to work with them is crucial for solving various mathematical problems. We will break down the expression, apply the properties of logarithms, and arrive at the final answer.

Understanding Logarithms

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A logarithm is the inverse operation of exponentiation. In other words, it is the power to which a base number must be raised to obtain a given value. For example, if we have the equation $2^5 = 32$, we can say that the logarithm of 32 to the base 2 is 5, denoted as $\log_2 32 = 5$. This means that 2 raised to the power of 5 equals 32.

Evaluating the Expression

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Now, let's focus on the given expression $\log _2\left(32^2\right)$. To evaluate this expression, we need to apply the properties of logarithms. One of the key properties is the power rule, which states that $\log_b (x^y) = y \log_b x$. This means that we can bring the exponent down and multiply it by the logarithm of the base.

Applying the Power Rule

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Using the power rule, we can rewrite the expression as:

$$\log _2\left(32^2\right) = 2 \log _2 32$$

Evaluating the Logarithm

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Now, we need to evaluate the logarithm $\log _2 32$. We know that $2^5 = 32$, so we can say that $\log _2 32 = 5$. Substituting this value into the expression, we get:

$$2 \log _2 32 = 2 \times 5 = 10$$

Conclusion

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In conclusion, we have evaluated the expression $\log _2\left(32^2\right)$ using the properties of logarithms. By applying the power rule and evaluating the logarithm, we arrived at the final answer of 10.

Final Answer

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The final answer is $\boxed{10}$.

Related Topics

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• Logarithmic properties
• Exponentiation
• Inverse operations

Further Reading

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• Khan Academy: Logarithms
• Math Is Fun: Logarithms
• Wolfram MathWorld: Logarithm

FAQs

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• Q: What is the logarithm of 32 to the base 2? A: The logarithm of 32 to the base 2 is 5.
• Q: How do you evaluate the expression $\log _2\left(32^2\right)$? A: You can apply the power rule and evaluate the logarithm to arrive at the final answer.
• Q: What is the final answer to the expression $\log _2\left(32^2\right)$? A: The final answer is 10.
  

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Introduction

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Logarithms are a fundamental concept in mathematics, and understanding how to work with them is crucial for solving various mathematical problems. In this article, we will address some of the most frequently asked questions about logarithms, providing clear and concise answers to help you better understand this important mathematical concept.

Q&A

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Q: What is a logarithm?

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A: A logarithm is the inverse operation of exponentiation. In other words, it is the power to which a base number must be raised to obtain a given value.

Q: What is the difference between a logarithm and an exponent?

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A: A logarithm is the inverse operation of an exponent. For example, if we have the equation $2^5 = 32$, we can say that the logarithm of 32 to the base 2 is 5, denoted as $\log_2 32 = 5$. This means that 2 raised to the power of 5 equals 32.

Q: How do you evaluate a logarithmic expression?

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A: To evaluate a logarithmic expression, you need to apply the properties of logarithms. One of the key properties is the power rule, which states that $\log_b (x^y) = y \log_b x$. This means that you can bring the exponent down and multiply it by the logarithm of the base.

Q: What is the power rule of logarithms?

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A: The power rule of logarithms states that $\log_b (x^y) = y \log_b x$. This means that you can bring the exponent down and multiply it by the logarithm of the base.

Q: How do you simplify a logarithmic expression?

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A: To simplify a logarithmic expression, you need to apply the properties of logarithms. You can use the power rule, the product rule, and the quotient rule to simplify the expression.

Q: What is the product rule of logarithms?

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A: The product rule of logarithms states that $\log_b (xy) = \log_b x + \log_b y$. This means that you can add the logarithms of two numbers to get the logarithm of their product.

Q: What is the quotient rule of logarithms?

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A: The quotient rule of logarithms states that $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$. This means that you can subtract the logarithm of a number from the logarithm of another number to get the logarithm of their quotient.

Q: How do you change the base of a logarithm?

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A: To change the base of a logarithm, you can use the change of base formula, which states that $\log_b x = \frac{\log_c x}{\log_c b}$. This means that you can change the base of a logarithm by dividing the logarithm of the number by the logarithm of the new base.

Q: What is the change of base formula?

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A: The change of base formula states that $\log_b x = \frac{\log_c x}{\log_c b}$. This means that you can change the base of a logarithm by dividing the logarithm of the number by the logarithm of the new base.

Conclusion

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In conclusion, logarithms are a fundamental concept in mathematics, and understanding how to work with them is crucial for solving various mathematical problems. By applying the properties of logarithms, you can simplify and evaluate logarithmic expressions. We hope that this article has provided you with a better understanding of logarithms and their applications.

Final Answer

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The final answer is $\boxed{10}$.

Related Topics

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• Logarithmic properties
• Exponentiation
• Inverse operations

Further Reading

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• Khan Academy: Logarithms
• Math Is Fun: Logarithms
• Wolfram MathWorld: Logarithm

FAQs

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• Q: What is the logarithm of 32 to the base 2? A: The logarithm of 32 to the base 2 is 5.
• Q: How do you evaluate the expression $\log _2\left(32^2\right)$? A: You can apply the power rule and evaluate the logarithm to arrive at the final answer.
• Q: What is the final answer to the expression $\log _2\left(32^2\right)$? A: The final answer is 10.