What Is The Value Of Log ⁡ 625 5 \log_{625} 5 Lo G 625 ​ 5 ?A. − 4 -4 − 4 B. − 1 4 -\frac{1}{4} − 4 1 ​ C. 1 4 \frac{1}{4} 4 1 ​ D. 4 4 4

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Introduction

In this article, we will explore the concept of logarithms and how to evaluate the value of $\log_{625} 5$. Logarithms are a fundamental concept in mathematics, and they have numerous applications in various fields, including science, engineering, and finance. Understanding logarithms is crucial for solving problems in these fields.

What are Logarithms?

A logarithm is the inverse operation of exponentiation. In other words, it is the power to which a base number must be raised to produce a given value. For example, if we have the equation $2^x = 8$, then the logarithm of 8 to the base 2 is 3, because $2^3 = 8$. This can be written as $\log_2 8 = 3$.

Evaluating the Value of $\log_{625} 5$

To evaluate the value of $\log_{625} 5$, we need to find the power to which 625 must be raised to produce 5. In other words, we need to find the exponent to which 625 must be raised to get 5.

Using the Change of Base Formula

One way to evaluate the value of $\log_{625} 5$ 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 $a$, $b$, and $c$ are positive real numbers and $c \neq 1$. We can use this formula to change the base of the logarithm from 625 to a more familiar base, such as 10.

Applying the Change of Base Formula

Using the change of base formula, we can write $\log_{625} 5 = \frac{\log_{10} 5}{\log_{10} 625}$. We can then use a calculator to evaluate the values of $\log_{10} 5$ and $\log_{10} 625$.

Evaluating the Numerator and Denominator

The value of $\log_{10} 5$ is approximately 0.69897. The value of $\log_{10} 625$ is approximately 2.7959.

Simplifying the Expression

Now that we have evaluated the numerator and denominator, we can simplify the expression. We have $\log_{625} 5 = \frac{0.69897}{2.7959}$.

Evaluating the Final Expression

Evaluating the final expression, we get $\log_{625} 5 = \frac{0.69897}{2.7959} \approx -0.25$.

Conclusion

In this article, we have explored the concept of logarithms and how to evaluate the value of $\log_{625} 5$. We have used the change of base formula to change the base of the logarithm from 625 to a more familiar base, such as 10. We have then evaluated the values of $\log_{10} 5$ and $\log_{10} 625$ using a calculator. Finally, we have simplified the expression and evaluated the final result.

Final Answer

The final answer is $\boxed{-\frac{1}{4}}$.

Discussion

The value of $\log_{625} 5$ is a fundamental concept in mathematics, and it has numerous applications in various fields. Understanding logarithms is crucial for solving problems in these fields. In this article, we have explored the concept of logarithms and how to evaluate the value of $\log_{625} 5$. We have used the change of base formula to change the base of the logarithm from 625 to a more familiar base, such as 10. We have then evaluated the values of $\log_{10} 5$ and $\log_{10} 625$ using a calculator. Finally, we have simplified the expression and evaluated the final result.

Related Topics

• Logarithmic functions
• Exponential functions
• Change of base formula
• Evaluating logarithms

References

• [1] "Logarithms" by Khan Academy
• [2] "Exponential and Logarithmic Functions" by Math Open Reference
• [3] "Change of Base Formula" by Wolfram MathWorld
  

Introduction

Logarithms are a fundamental concept in mathematics, and they have numerous applications in various fields. However, many people find logarithms to be confusing and difficult to understand. In this article, we will answer some of the most frequently asked questions about logarithms.

Q: What is a logarithm?

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 produce a given value.

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

A: A logarithm is the inverse operation of an exponent. For example, if we have the equation $2^x = 8$, then the logarithm of 8 to the base 2 is 3, because $2^3 = 8$. This can be written as $\log_2 8 = 3$.

Q: How do I evaluate a logarithm?

A: To evaluate a logarithm, you need to find the power to which the base number must be raised to produce the given value. You can use the change of base formula to change the base of the logarithm to a more familiar base, such as 10.

Q: What is the change of base formula?

A: The change of base formula states that $\log_b a = \frac{\log_c a}{\log_c b}$, where $a$, $b$, and $c$ are positive real numbers and $c \neq 1$. This formula allows you to change the base of the logarithm to a more familiar base.

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

A: To use the change of base formula, you need to identify the base, the value, and the new base. Then, you can plug these values into the formula and simplify.

Q: What is the logarithm of 1?

A: The logarithm of 1 is 0, because any number raised to the power of 0 is 1.

Q: What is the logarithm of 0?

A: The logarithm of 0 is undefined, because any number raised to a negative power is not defined.

Q: Can I have a negative base?

A: No, you cannot have a negative base. The base of a logarithm must be a positive real number.

Q: Can I have a base of 1?

A: No, you cannot have a base of 1. The base of a logarithm must be a positive real number other than 1.

Q: What is the logarithm of a negative number?

A: The logarithm of a negative number is undefined, because any number raised to a power is not defined for negative numbers.

Q: What is the logarithm of a complex number?

A: The logarithm of a complex number is a complex number, because complex numbers can be raised to powers.

Q: Can I use a calculator to evaluate logarithms?

A: Yes, you can use a calculator to evaluate logarithms. Most calculators have a built-in logarithm function that you can use to evaluate logarithms.

Q: What is the logarithm of e?

A: The logarithm of e is 1, because e is the base of the natural logarithm.

Q: What is the logarithm of pi?

A: The logarithm of pi is approximately 0.49715, because pi is approximately 3.14159.

Conclusion

In this article, we have answered some of the most frequently asked questions about logarithms. We have covered topics such as the definition of a logarithm, the difference between a logarithm and an exponent, and how to evaluate a logarithm. We have also covered topics such as the change of base formula, the logarithm of 1, and the logarithm of a negative number.

Final Answer

The final answer is that logarithms are a fundamental concept in mathematics, and they have numerous applications in various fields. Understanding logarithms is crucial for solving problems in these fields.

Related Topics

• Logarithmic functions
• Exponential functions
• Change of base formula
• Evaluating logarithms

References

• [1] "Logarithms" by Khan Academy
• [2] "Exponential and Logarithmic Functions" by Math Open Reference
• [3] "Change of Base Formula" by Wolfram MathWorld