Evaluate: Log ⁡ 64 128 \log _{64} 128 Lo G 64 ​ 128

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Introduction

In mathematics, logarithms are a fundamental concept that helps us understand the relationship between numbers and their powers. The logarithm of a number to a certain base is the exponent to which the base must be raised to produce the number. In this article, we will evaluate the logarithm $\log _{64} 128$, which is a common problem in mathematics.

Understanding Logarithms

Before we dive into the evaluation of $\log _{64} 128$, let's understand the concept of logarithms. A logarithm is the inverse operation of exponentiation. In other words, if we have a number $x$ and a base $b$, then the logarithm of $x$ to the base $b$ is the exponent to which $b$ must be raised to produce $x$. For example, if we have $\log _{2} 8$, then we are looking for the exponent to which 2 must be raised to produce 8. Since $2^3 = 8$, we can say that $\log _{2} 8 = 3$.

Evaluating $\log _{64} 128$

Now that we have a good understanding of logarithms, let's evaluate $\log _{64} 128$. To do this, we need to find the exponent to which 64 must be raised to produce 128. Since 64 is a power of 2, we can rewrite it as $2^6$. Therefore, we can rewrite $\log _{64} 128$ as $\log _{2^6} 128$.

Using Properties of Logarithms

We can use the property of logarithms that states $\log _{a^b} c = \frac{1}{b} \log _{a} c$. Applying this property to our problem, we get:

$$\log _{2^6} 128 = \frac{1}{6} \log _{2} 128$$

Evaluating $\log _{2} 128$

Now that we have simplified the expression, we need to evaluate $\log _{2} 128$. To do this, we can use the fact that $128 = 2^7$. Therefore, we can rewrite $\log _{2} 128$ as $\log _{2} 2^7$.

Using Properties of Logarithms Again

We can use the property of logarithms that states $\log _{a} a^b = b$. Applying this property to our problem, we get:

$$\log _{2} 2^7 = 7$$

Putting it All Together

Now that we have evaluated $\log _{2} 128$, we can substitute this value back into our original expression:

$$\log _{2^6} 128 = \frac{1}{6} \log _{2} 128 = \frac{1}{6} \cdot 7 = \frac{7}{6}$$

Conclusion

In this article, we evaluated the logarithm $\log _{64} 128$. We used the properties of logarithms to simplify the expression and eventually arrived at the answer $\frac{7}{6}$. This problem is a great example of how logarithms can be used to solve problems involving exponents and powers.

Frequently Asked Questions

• What is the logarithm of 128 to the base 64?
• How do you evaluate a logarithm with a base that is a power of another number?
• What is the relationship between logarithms and exponents?

Answers

• The logarithm of 128 to the base 64 is $\frac{7}{6}$.
• To evaluate a logarithm with a base that is a power of another number, you can use the property of logarithms that states $\log _{a^b} c = \frac{1}{b} \log _{a} c$.
• The relationship between logarithms and exponents is that logarithms are the inverse operation of exponentiation.

Further Reading

If you want to learn more about logarithms and how to evaluate them, here are some additional resources:

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

Conclusion

In conclusion, evaluating logarithms can be a challenging but rewarding task. By using the properties of logarithms and simplifying expressions, we can arrive at the correct answer. We hope this article has been helpful in understanding how to evaluate logarithms and has provided you with a better understanding of this important mathematical concept.

Introduction

Logarithms are a fundamental concept in mathematics that can be used to solve a wide range of problems. However, they can also be confusing and difficult to understand, especially for those who are new to the subject. In this article, we will answer some of the most frequently asked questions about logarithms, covering topics such as the definition of a logarithm, how to evaluate logarithms, and common logarithmic identities.

Q: What is a logarithm?

A: A logarithm is the inverse operation of exponentiation. In other words, if we have a number $x$ and a base $b$, then the logarithm of $x$ to the base $b$ is the exponent to which $b$ must be raised to produce $x$. For example, if we have $\log _{2} 8$, then we are looking for the exponent to which 2 must be raised to produce 8. Since $2^3 = 8$, we can say that $\log _{2} 8 = 3$.

Q: How do I evaluate a logarithm?

A: To evaluate a logarithm, you need to find the exponent to which the base must be raised to produce the given number. For example, if we have $\log _{2} 16$, then we need to find the exponent to which 2 must be raised to produce 16. Since $2^4 = 16$, we can say that $\log _{2} 16 = 4$.

Q: What is the relationship between logarithms and exponents?

A: Logarithms and exponents are inverse operations. In other words, if we have a logarithm $\log _{a} x$, then we can rewrite it as an exponent $a^{\log _{a} x}$. For example, if we have $\log _{2} 8$, then we can rewrite it as $2^{\log _{2} 8}$. Since $2^3 = 8$, we can say that $\log _{2} 8 = 3$.

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

A: A logarithm is the inverse operation of an exponent. In other words, if we have a number $x$ and a base $b$, then the logarithm of $x$ to the base $b$ is the exponent to which $b$ must be raised to produce $x$. An exponent, on the other hand, is the power to which a number must be raised to produce a given value.

Q: How do I use logarithmic identities to simplify expressions?

A: Logarithmic identities are formulas that can be used to simplify expressions involving logarithms. For example, the identity $\log _{a} (bc) = \log _{a} b + \log _{a} c$ can be used to simplify expressions involving the product of two numbers. Similarly, the identity $\log _{a} \left(\frac{b}{c}\right) = \log _{a} b - \log _{a} c$ can be used to simplify expressions involving the quotient of two numbers.

Q: What is the change of base formula?

A: The change of base formula is a formula that can be used to change the base of a logarithm. The formula is $\log _{a} x = \frac{\log _{b} x}{\log _{b} a}$, where $a$ and $b$ are any positive numbers.

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

A: To use the change of base formula, you need to choose a new base $b$ and then rewrite the logarithm in terms of the new base. For example, if we have $\log _{2} 8$, then we can use the change of base formula to rewrite it as $\frac{\log _{10} 8}{\log _{10} 2}$.

Q: What is the logarithmic identity for the sum of two logarithms?

A: The logarithmic identity for the sum of two logarithms is $\log _{a} (bc) = \log _{a} b + \log _{a} c$.

Q: What is the logarithmic identity for the difference of two logarithms?

A: The logarithmic identity for the difference of two logarithms is $\log _{a} \left(\frac{b}{c}\right) = \log _{a} b - \log _{a} c$.

Q: What is the logarithmic identity for the product of two logarithms?

A: The logarithmic identity for the product of two logarithms is $\log _{a} (bc) = \log _{a} b + \log _{a} c$.

Q: What is the logarithmic identity for the quotient of two logarithms?

A: The logarithmic identity for the quotient of two logarithms is $\log _{a} \left(\frac{b}{c}\right) = \log _{a} b - \log _{a} c$.

Conclusion

In this article, we have answered some of the most frequently asked questions about logarithms, covering topics such as the definition of a logarithm, how to evaluate logarithms, and common logarithmic identities. We hope that this article has been helpful in understanding logarithms and has provided you with a better understanding of this important mathematical concept.

Further Reading

If you want to learn more about logarithms and how to evaluate them, here are some additional resources:

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

Conclusion

In conclusion, logarithms are a fundamental concept in mathematics that can be used to solve a wide range of problems. By understanding the definition of a logarithm, how to evaluate logarithms, and common logarithmic identities, you can become proficient in using logarithms to simplify expressions and solve problems. We hope that this article has been helpful in understanding logarithms and has provided you with a better understanding of this important mathematical concept.