Evaluate: Log ⁡ 16 64 \log_{16} 64 Lo G 16 ​ 64 A. 1 3 \frac{1}{3} 3 1 ​ B. 2 3 \frac{2}{3} 3 2 ​ C. 3 2 \frac{3}{2} 2 3 ​

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Introduction

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Logarithmic expressions are a fundamental concept in mathematics, and understanding how to evaluate them is crucial for solving various mathematical problems. In this article, we will focus on evaluating the logarithmic expression $\log_{16} 64$. We will break down the problem step by step, explaining the concepts and formulas involved.

What is a Logarithm?

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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 $2^x = 8$, then the logarithm of 8 to the base 2 is equal to x, which is 3. This can be written as $\log_2 8 = 3$.

Evaluating $\log_{16} 64$

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To evaluate the logarithmic expression $\log_{16} 64$, we need to find the power to which 16 must be raised to obtain 64. In other words, we need to find the value of x in the equation $16^x = 64$.

Using the Change of Base Formula

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One way to evaluate the logarithmic expression is to use the change of base formula. The change of base formula states that $\log_b a = \frac{\log_c a}{\log_c b}$, where c is any positive real number not equal to 1. We can use this formula to rewrite the logarithmic expression in terms of a more familiar base, such as the common logarithm (base 10) or the natural logarithm (base e).

Applying the Change of Base Formula

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Using the change of base formula, we can rewrite the logarithmic expression as follows:

$$\log_{16} 64 = \frac{\log 64}{\log 16}$$

Simplifying the Expression

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We can simplify the expression by using the properties of logarithms. Specifically, we can use the fact that $\log a^b = b \log a$ to rewrite the expression as follows:

$$\frac{\log 64}{\log 16} = \frac{\log 2^6}{\log 2^4}$$

Evaluating the Expression

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We can evaluate the expression by simplifying the logarithms:

$$\frac{\log 2^6}{\log 2^4} = \frac{6 \log 2}{4 \log 2} = \frac{6}{4} = \frac{3}{2}$$

Conclusion

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In conclusion, the value of the logarithmic expression $\log_{16} 64$ is $\frac{3}{2}$. We used the change of base formula and the properties of logarithms to simplify the expression and evaluate it.

Frequently Asked Questions

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• Q: What is the logarithm of 64 to the base 16? A: The logarithm of 64 to the base 16 is $\frac{3}{2}$.
• Q: How do I evaluate a logarithmic expression? A: To evaluate a logarithmic expression, you can use the change of base formula and the properties of logarithms.
• Q: What is the change of base formula? A: The change of base formula is $\log_b a = \frac{\log_c a}{\log_c b}$, where c is any positive real number not equal to 1.

Final Thoughts

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Evaluating logarithmic expressions is an important skill in mathematics, and it requires a good understanding of the concepts and formulas involved. In this article, we used the change of base formula and the properties of logarithms to evaluate the logarithmic expression $\log_{16} 64$. We hope that this article has provided you with a better understanding of logarithmic expressions and how to evaluate them.

Additional Resources

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

References

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• "Logarithms" by Khan Academy
• "Logarithms" by Math Is Fun
• "Logarithm" by Wolfram MathWorld
  

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Introduction

---

Logarithmic expressions are a fundamental concept in mathematics, and understanding how to evaluate them is crucial for solving various mathematical problems. In this article, we will provide a Q&A guide to help you better understand logarithmic expressions and how to evaluate them.

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: How do I evaluate a logarithmic expression?

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A: To evaluate a logarithmic expression, you can use the change of base formula and the properties of logarithms. The change of base formula is $\log_b a = \frac{\log_c a}{\log_c b}$, where c is any positive real number not equal to 1.

Q: What is the change of base formula?

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A: The change of base formula is $\log_b a = \frac{\log_c a}{\log_c b}$, where c is any positive real number not equal to 1. This formula allows you to rewrite a logarithmic expression in terms of a more familiar base, such as the common logarithm (base 10) or the natural logarithm (base e).

Q: How do I use the change of base formula?

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A: To use the change of base formula, you need to identify the base and the argument of the logarithmic expression. Then, you can rewrite the expression using the formula. For example, if you have the expression $\log_{16} 64$, you can rewrite it as $\frac{\log 64}{\log 16}$.

Q: What are the properties of logarithms?

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A: The properties of logarithms are:

• $\log a^b = b \log a$
• $\log \frac{a}{b} = \log a - \log b$
• $\log (ab) = \log a + \log b$

Q: How do I use the properties of logarithms?

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A: To use the properties of logarithms, you need to identify the expression and apply the appropriate property. For example, if you have the expression $\log 2^6$, you can use the property $\log a^b = b \log a$ to rewrite it as $6 \log 2$.

Q: What is the difference between a logarithmic expression and an exponential expression?

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A: A logarithmic expression is the inverse of an exponential expression. In other words, if you have an exponential expression $a^b$, the corresponding logarithmic expression is $\log_a b$. For example, if you have the expression $2^3$, the corresponding logarithmic expression is $\log_2 3$.

Q: How do I evaluate a logarithmic expression with a negative base?

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A: To evaluate a logarithmic expression with a negative base, you need to use the property $\log (-a) = \log a + i \pi$, where $i$ is the imaginary unit. For example, if you have the expression $\log (-2)$, you can rewrite it as $\log 2 + i \pi$.

Q: What is the relationship between logarithmic expressions and trigonometric functions?

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A: Logarithmic expressions and trigonometric functions are related through the use of the logarithmic and exponential functions. For example, the natural logarithm (base e) is related to the exponential function $e^x$.

Q: How do I use logarithmic expressions in real-world applications?

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A: Logarithmic expressions are used in a variety of real-world applications, including:

• Finance: Logarithmic expressions are used to calculate interest rates and investment returns.
• Science: Logarithmic expressions are used to calculate the pH of a solution and the concentration of a substance.
• Engineering: Logarithmic expressions are used to calculate the power and energy of a system.

Conclusion

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In conclusion, logarithmic expressions are a fundamental concept in mathematics, and understanding how to evaluate them is crucial for solving various mathematical problems. We hope that this Q&A guide has provided you with a better understanding of logarithmic expressions and how to evaluate them.

Additional Resources

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

References

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• "Logarithms" by Khan Academy
• "Logarithms" by Math Is Fun
• "Logarithm" by Wolfram MathWorld