Evaluate $\log _6 40$.

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Introduction

In this article, we will delve into the world of logarithms and explore the concept of evaluating a logarithmic expression. Specifically, we will focus on evaluating the expression $\log _6 40$. This involves understanding the properties of logarithms, the change of base formula, and how to apply these concepts to solve the given problem.

Understanding 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 obtain a given value. For example, if we have the expression $\log _2 8$, it means that we need to find the power to which 2 must be raised to get 8. In this case, the answer is 3, because $2^3 = 8$. This is the fundamental concept of logarithms, and it is essential to grasp this idea to evaluate logarithmic expressions.

Change of Base Formula

The change of base formula is a powerful tool that allows us to evaluate logarithmic expressions with different bases. The formula states that $\log _a b = \frac{\log _c b}{\log _c a}$, where $a$, $b$, and $c$ are positive real numbers and $c \neq 1$. This formula enables us to change the base of a logarithm to a more convenient one, making it easier to evaluate the expression.

Evaluating $\log _6 40$

To evaluate $\log _6 40$, we can use the change of base formula. We can choose a convenient base, such as 10, and rewrite the expression as $\frac{\log _{10} 40}{\log _{10} 6}$. This allows us to use the logarithm of 40 and 6 with base 10, which is a common and easily accessible base.

Calculating Logarithms

To calculate the logarithms in the expression, we can use a calculator or a logarithmic table. For example, if we use a calculator, we can find that $\log _{10} 40 \approx 1.60206$ and $\log _{10} 6 \approx 0.77815$. These values can be used to evaluate the expression.

Evaluating the Expression

Now that we have the values of the logarithms, we can substitute them into the expression and evaluate it. We have $\frac{\log _{10} 40}{\log _{10} 6} \approx \frac{1.60206}{0.77815} \approx 2.065$. Therefore, the value of $\log _6 40$ is approximately 2.065.

Conclusion

In this article, we evaluated the logarithmic expression $\log _6 40$ using the change of base formula. We chose a convenient base, calculated the logarithms, and substituted the values into the expression to obtain the final answer. This demonstrates the power of logarithms and the importance of understanding their properties and applications.

Final Answer

The final answer to the problem is $\boxed{2.065}$.

Additional Resources

For those who want to learn more about logarithms and their applications, here are some additional resources:

• Logarithm Wikipedia Page
• Logarithm Calculator
• Logarithm Table

Related Problems

If you want to practice evaluating logarithmic expressions, here are some related problems:

• Evaluate $\log _3 27$
• Evaluate $\log _2 16$
• Evaluate $\log _4 64$

These problems can help you reinforce your understanding of logarithms and their applications.

Introduction

In our previous article, we explored the concept of logarithms and evaluated the expression $\log _6 40$. However, we understand that there may be many more questions and doubts that readers may have. 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.

Q1: What is the difference between a logarithm and an exponent?

A1: 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 expression $\log _2 8$, it means that we need to find the power to which 2 must be raised to get 8. In this case, the answer is 3, because $2^3 = 8$.

Q2: What is the change of base formula?

A2: The change of base formula is a powerful tool that allows us to evaluate logarithmic expressions with different bases. The formula states that $\log _a b = \frac{\log _c b}{\log _c a}$, where $a$, $b$, and $c$ are positive real numbers and $c \neq 1$. This formula enables us to change the base of a logarithm to a more convenient one, making it easier to evaluate the expression.

Q3: How do I evaluate a logarithmic expression?

A3: To evaluate a logarithmic expression, you need to follow these steps:

1. Identify the base and the argument of the logarithm.
2. Choose a convenient base to change the base of the logarithm.
3. Calculate the logarithms using the chosen base.
4. Substitute the values into the expression and evaluate it.

Q4: What is the logarithm of 1?

A4: The logarithm of 1 is 0, regardless of the base. This is because any number raised to the power of 0 is equal to 1.

Q5: What is the logarithm of 0?

A5: The logarithm of 0 is undefined, regardless of the base. This is because any number raised to a negative power is undefined.

Q6: Can I use a calculator to evaluate logarithmic expressions?

A6: Yes, you can use a calculator to evaluate logarithmic expressions. Most calculators have a built-in logarithm function that allows you to enter the base and the argument of the logarithm and obtain the result.

Q7: How do I convert a logarithmic expression to exponential form?

A7: To convert a logarithmic expression to exponential form, you need to follow these steps:

1. Identify the base and the argument of the logarithm.
2. Raise the base to the power of the argument to obtain the exponential form.

Q8: What is the relationship between logarithms and exponents?

A8: Logarithms and exponents are inverse operations. In other words, they are two sides of the same coin. Logarithms are used to find the power to which a base number must be raised to obtain a given value, while exponents are used to find the result of raising a base number to a given power.

Q9: Can I use logarithms to solve equations?

A9: Yes, you can use logarithms to solve equations. Logarithms can be used to simplify equations and make them easier to solve.

Q10: What are some real-world applications of logarithms?

A10: Logarithms have many real-world applications, including:

• Finance: Logarithms are used to calculate interest rates and investment returns.
• Science: Logarithms are used to calculate the pH of a solution and the concentration of a substance.
• Engineering: Logarithms are used to calculate the gain of an amplifier and the frequency response of a system.

Conclusion

In this article, we addressed some of the most frequently asked questions about logarithms, providing clear and concise answers to help you better understand this important mathematical concept. We hope that this article has been helpful in clarifying any doubts you may have had about logarithms.

Final Answer

The final answer to the problem is $\boxed{2.065}$.

Additional Resources

For those who want to learn more about logarithms and their applications, here are some additional resources:

• Logarithm Wikipedia Page
• Logarithm Calculator
• Logarithm Table

Related Problems

If you want to practice evaluating logarithmic expressions, here are some related problems:

• Evaluate $\log _3 27$
• Evaluate $\log _2 16$
• Evaluate $\log _4 64$

These problems can help you reinforce your understanding of logarithms and their applications.