Evaluate \[$\log_{1/2} 8\$\].

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Introduction

In mathematics, logarithms are a fundamental concept that helps us solve equations and express complex relationships in a simpler form. The logarithm of a number to a certain base is the exponent to which the base must be raised to produce that number. In this article, we will evaluate the expression $\log_{1/2} 8$, which involves finding the logarithm of 8 to the base 1/2.

Understanding Logarithms

Before we dive into the evaluation of $\log_{1/2} 8$, let's briefly review the concept of logarithms. The logarithm of a number $x$ to a base $b$ is denoted by $\log_b x$ and is defined as the exponent to which $b$ must be raised to produce $x$. In other words, if $b^y = x$, then $\log_b x = y$. For example, $\log_{10} 100 = 2$ because $10^2 = 100$.

Properties of Logarithms

Logarithms have several important properties that we will use to evaluate $\log_{1/2} 8$. One of the most useful properties is the change of base formula, which states that $\log_b x = \frac{\log_c x}{\log_c b}$ for any positive real numbers $b$, $c$, and $x$. This formula allows us to change the base of a logarithm to any other base.

Evaluating $\log_{1/2} 8$

To evaluate $\log_{1/2} 8$, we can use the change of base formula. Let's change the base to 2, which is the same as the base of the logarithm we are trying to evaluate. We have:

$$\log_{1/2} 8 = \frac{\log_2 8}{\log_2 (1/2)}$$

Simplifying the Expression

Now, let's simplify the expression inside the logarithms. We have:

$$\log_2 8 = 3$$

because $2^3 = 8$. We also have:

$$\log_2 (1/2) = -1$$

because $2^{-1} = 1/2$.

Substituting the Values

Now, let's substitute the values we found into the expression:

$$\log_{1/2} 8 = \frac{3}{-1} = -3$$

Conclusion

In this article, we evaluated the expression $\log_{1/2} 8$ using the change of base formula and properties of logarithms. We found that $\log_{1/2} 8 = -3$. This result is consistent with the definition of logarithms, which states that $\log_b x = y$ if $b^y = x$. We hope this article has helped you understand the concept of logarithms and how to evaluate expressions involving logarithms.

Additional Examples

Here are a few more examples of evaluating logarithmic expressions:

• $\log_{1/3} 27 = \frac{\log_3 27}{\log_3 (1/3)} = \frac{3}{-1} = -3$
• $\log_{1/4} 16 = \frac{\log_4 16}{\log_4 (1/4)} = \frac{2}{-1} = -2$
• $\log_{1/5} 25 = \frac{\log_5 25}{\log_5 (1/5)} = \frac{2}{-1} = -2$

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 express the magnitude of earthquakes and the brightness of stars.
• Engineering: Logarithms are used to design electronic circuits and calculate signal strengths.

Final Thoughts

In conclusion, logarithms are a powerful tool for solving equations and expressing complex relationships in a simpler form. By understanding the properties of logarithms and how to evaluate expressions involving logarithms, we can solve a wide range of problems in mathematics, science, and engineering. We hope this article has helped you understand the concept of logarithms and how to apply them in real-world situations.

Introduction

In our previous article, we evaluated the expression $\log_{1/2} 8$ using the change of base formula and properties of logarithms. In this article, we will answer some frequently asked questions about logarithm evaluation.

Q: What is the change of base formula?

A: The change of base formula is a property of logarithms that allows us to change the base of a logarithm to any other base. It is given by:

$$\log_b x = \frac{\log_c x}{\log_c b}$$

for any positive real numbers $b$, $c$, and $x$.

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

A: To apply the change of base formula, you need to:

1. Identify the base of the logarithm you want to evaluate.
2. Choose a new base to which you want to change the logarithm.
3. Use the formula to rewrite the logarithm in terms of the new base.

Q: What are some common logarithmic expressions that can be evaluated using the change of base formula?

A: Some common logarithmic expressions that can be evaluated using the change of base formula include:

• $\log_{1/2} 8$
• $\log_{1/3} 27$
• $\log_{1/4} 16$
• $\log_{1/5} 25$

Q: How do I simplify logarithmic expressions?

A: To simplify logarithmic expressions, you can use the following properties:

• $\log_b 1 = 0$
• $\log_b b = 1$
• $\log_b (x \cdot y) = \log_b x + \log_b y$
• $\log_b (x / y) = \log_b x - \log_b y$

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

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

• Finance: Logarithms are used to calculate interest rates and investment returns.
• Science: Logarithms are used to express the magnitude of earthquakes and the brightness of stars.
• Engineering: Logarithms are used to design electronic circuits and calculate signal strengths.

Q: How do I evaluate logarithmic expressions with negative bases?

A: To evaluate logarithmic expressions with negative bases, you can use the following property:

$$\log_{-b} x = \log_{b} (-x)$$

Q: What are some common mistakes to avoid when evaluating logarithmic expressions?

A: Some common mistakes to avoid when evaluating logarithmic expressions include:

• Forgetting to change the base: Make sure to change the base of the logarithm to the new base.
• Not simplifying the expression: Simplify the expression as much as possible before evaluating it.
• Not using the correct properties: Use the correct properties of logarithms to simplify the expression.

Q: How do I evaluate logarithmic expressions with fractional bases?

A: To evaluate logarithmic expressions with fractional bases, you can use the following property:

$$\log_{b/m} x = \frac{\log_b x}{\log_b (m)}$$

Q: What are some advanced topics in logarithm evaluation?

A: Some advanced topics in logarithm evaluation include:

• Logarithmic identities: Logarithmic identities are equations that involve logarithms and can be used to simplify expressions.
• Logarithmic inequalities: Logarithmic inequalities are inequalities that involve logarithms and can be used to solve problems.
• Logarithmic equations: Logarithmic equations are equations that involve logarithms and can be used to solve problems.

Conclusion

In this article, we answered some frequently asked questions about logarithm evaluation. We hope this article has helped you understand the concept of logarithms and how to apply them in real-world situations.