Evaluate: Log ⁡ 128 4 \log _{128} 4 Lo G 128 ​ 4

Introduction

In mathematics, logarithms are a fundamental concept that helps us understand the relationship between numbers and their exponents. 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 $\log _{128} 4$ and explore its properties.

Understanding 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 $x$. This can be represented mathematically as:

$$\log _{b} x = y \iff b^y = x$$

For example, if we want to find the logarithm of 16 to the base 2, we can ask ourselves what exponent we need to raise 2 to in order to get 16. The answer is 4, because $2^4 = 16$. Therefore, $\log _{2} 16 = 4$.

Evaluating $\log _{128} 4$

To evaluate $\log _{128} 4$, we need to find the exponent to which 128 must be raised to produce 4. One way to approach this is to use the fact that $128 = 2^7$ and $4 = 2^2$. We can then rewrite the logarithm as:

$$\log _{128} 4 = \log _{2^7} 2^2$$

Using the property of logarithms that $\log _{b^c} x = \frac{1}{c} \log _{b} x$, we can simplify this expression to:

$$\log _{128} 4 = \frac{1}{7} \log _{2} 2^2$$

Simplifying the Expression

Now we can simplify the expression further by using the fact that $\log _{b} b^x = x$. This gives us:

$$\log _{128} 4 = \frac{1}{7} \cdot 2$$

Final Answer

Therefore, the final answer to the logarithm $\log _{128} 4$ is $\frac{2}{7}$.

Conclusion

In this article, we evaluated the logarithm $\log _{128} 4$ and explored its properties. We used the fact that $128 = 2^7$ and $4 = 2^2$ to rewrite the logarithm in a simpler form, and then used the property of logarithms that $\log _{b^c} x = \frac{1}{c} \log _{b} x$ to simplify the expression. Finally, we used the fact that $\log _{b} b^x = x$ to arrive at the final answer of $\frac{2}{7}$.

Properties of Logarithms

Logarithms have several important properties that make them useful in mathematics and other fields. Some of these properties include:

• The product rule: $\log _{b} (xy) = \log _{b} x + \log _{b} y$
• The quotient rule: $\log _{b} \frac{x}{y} = \log _{b} x - \log _{b} y$
• The power rule: $\log _{b} x^y = y \log _{b} x$
• The change of base formula: $\log _{b} x = \frac{\log _{c} x}{\log _{c} b}$

These properties can be used to simplify and evaluate logarithmic expressions, and are essential for working with logarithms in mathematics and other fields.

Applications of Logarithms

Logarithms have many practical applications in mathematics, science, and engineering. Some examples include:

• Finance: Logarithms are used to calculate interest rates and investment returns.
• Science: Logarithms are used to measure the intensity of earthquakes and the brightness of stars.
• Engineering: Logarithms are used to design and optimize systems, such as electronic circuits and communication networks.
• Computer Science: Logarithms are used in algorithms for searching and sorting data, and in the design of data structures such as hash tables.

Conclusion

In conclusion, logarithms are a fundamental concept in mathematics that have many practical applications in science, engineering, and other fields. In this article, we evaluated the logarithm $\log _{128} 4$ and explored its properties, using the fact that $128 = 2^7$ and $4 = 2^2$ to rewrite the logarithm in a simpler form. We also discussed the properties and applications of logarithms, and provided examples of how they are used in mathematics and other fields.

Introduction

Logarithms can be a complex and confusing topic, especially for those who are new to mathematics. 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 logarithmic expressions, and the properties of logarithms.

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 $x$. This can be represented mathematically as:

$$\log _{b} x = y \iff b^y = x$$

Q: How do I evaluate a logarithmic expression?

A: To evaluate a logarithmic expression, you need to find the exponent to which the base must be raised to produce the number. For example, to evaluate $\log _{2} 16$, you need to find the exponent to which 2 must be raised to produce 16. The answer is 4, because $2^4 = 16$. Therefore, $\log _{2} 16 = 4$.

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

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 $x$. An exponent, on the other hand, is the power to which a number is raised. For example, $2^3$ is an exponent, while $\log _{2} 8$ is a logarithm.

Q: What are the properties of logarithms?

A: Logarithms have several important properties that make them useful in mathematics and other fields. Some of these properties include:

• The product rule: $\log _{b} (xy) = \log _{b} x + \log _{b} y$
• The quotient rule: $\log _{b} \frac{x}{y} = \log _{b} x - \log _{b} y$
• The power rule: $\log _{b} x^y = y \log _{b} x$
• The change of base formula: $\log _{b} x = \frac{\log _{c} x}{\log _{c} b}$

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

A: The change of base formula is used to change the base of a logarithm from one base to another. For example, if we want to find the logarithm of 16 to the base 2, but we only know the logarithm of 16 to the base 10, we can use the change of base formula to find the answer. The formula is:

$$\log _{b} x = \frac{\log _{c} x}{\log _{c} b}$$

Q: What are some common logarithmic expressions?

A: Some common logarithmic expressions include:

• $\log _{2} 2^x$
• $\log _{10} 10^x$
• $\log _{e} e^x$
• $\log _{b} b^x$

Q: How do I use logarithms in real-life situations?

A: Logarithms have many practical applications in mathematics, science, and engineering. Some examples include:

• Finance: Logarithms are used to calculate interest rates and investment returns.
• Science: Logarithms are used to measure the intensity of earthquakes and the brightness of stars.
• Engineering: Logarithms are used to design and optimize systems, such as electronic circuits and communication networks.
• Computer Science: Logarithms are used in algorithms for searching and sorting data, and in the design of data structures such as hash tables.

Conclusion

In conclusion, logarithms are a fundamental concept in mathematics that have many practical applications in science, engineering, and other fields. In this article, we answered some of the most frequently asked questions about logarithms, covering topics such as the definition of a logarithm, how to evaluate logarithmic expressions, and the properties of logarithms. We hope that this article has been helpful in understanding logarithms and how they are used in real-life situations.