Which Is Equivalent To Log ⁡ 1 2 462 \log _{\frac{1}{2}} 462 Lo G 2 1 ​ ​ 462 ?A. Log ⁡ 1 2 462 Log ⁡ 462 1 2 \frac{\log _{\frac{1}{2}} 462}{\log _{462} \frac{1}{2}} L O G 462 ​ 2 1 ​ L O G 2 1 ​ ​ 462 ​ B. Log ⁡ 462 Log ⁡ 1 2 \frac{\log 462}{\log \frac{1}{2}} L O G 2 1 ​ L O G 462 ​ C. Log ⁡ 1 2 Log ⁡ 462 \frac{\log \frac{1}{2}}{\log 462} L O G 462 L O G 2 1 ​ ​

Introduction

Logarithmic functions are a fundamental concept in mathematics, and they have numerous applications in various fields, including science, engineering, and finance. In this article, we will explore the concept of logarithmic equivalents and how to determine the equivalent of a given logarithmic expression.

What are Logarithmic Equivalents?

Logarithmic equivalents refer to the process of converting a logarithmic expression from one base to another. This is often necessary when working with different bases or when simplifying complex logarithmic expressions.

The Change of Base Formula

The change of base formula is a fundamental concept in logarithmic equivalents. It states that:

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

where $a$, $b$, and $c$ are positive real numbers, and $c \neq 1$.

Applying the Change of Base Formula

To determine the equivalent of $\log _{\frac{1}{2}} 462$, we can apply the change of base formula. Let's choose base 10 as our new base.

$$\log _{\frac{1}{2}} 462 = \frac{\log_{10} 462}{\log_{10} \frac{1}{2}}$$

Simplifying the Expression

Now, let's simplify the expression by evaluating the logarithms.

$$\log_{10} 462 \approx 2.663$$

$$\log_{10} \frac{1}{2} \approx -0.301$$

Substituting these values into the expression, we get:

$$\frac{\log_{10} 462}{\log_{10} \frac{1}{2}} \approx \frac{2.663}{-0.301} \approx -8.84$$

Comparing the Options

Now, let's compare the result with the given options.

A. $\frac{\log _{\frac{1}{2}} 462}{\log _{462} \frac{1}{2}}$

B. $\frac{\log 462}{\log \frac{1}{2}}$

C. $\frac{\log \frac{1}{2}}{\log 462}$

Based on our calculation, option A is the correct equivalent of $\log _{\frac{1}{2}} 462$.

Conclusion

In conclusion, logarithmic equivalents are an essential concept in mathematics, and the change of base formula is a powerful tool for converting logarithmic expressions from one base to another. By applying the change of base formula and simplifying the expression, we can determine the equivalent of a given logarithmic expression.

Common Logarithms

Common logarithms are logarithms with base 10. They are denoted by $\log x$ and are used extensively in mathematics and science.

Natural Logarithms

Natural logarithms are logarithms with base $e$. They are denoted by $\ln x$ and are used extensively in mathematics and science.

Properties of Logarithms

Logarithms have several important properties, including:

• Product Rule: $\log (ab) = \log a + \log b$
• Quotient Rule: $\log \left(\frac{a}{b}\right) = \log a - \log b$
• Power Rule: $\log a^b = b \log a$

Solving Logarithmic Equations

Logarithmic equations are equations that involve logarithmic expressions. They can be solved using the properties of logarithms and the change of base formula.

Example 1

Solve the equation $\log_2 x = 3$.

Using the change of base formula, we can rewrite the equation as:

$$\frac{\log_{10} x}{\log_{10} 2} = 3$$

Multiplying both sides by $\log_{10} 2$, we get:

$$\log_{10} x = 3 \log_{10} 2$$

Using the power rule, we can rewrite the equation as:

$$\log_{10} x = \log_{10} 2^3$$

Therefore, $x = 2^3 = 8$.

Example 2

Solve the equation $\log_3 x = 2$.

Using the change of base formula, we can rewrite the equation as:

$$\frac{\log_{10} x}{\log_{10} 3} = 2$$

Multiplying both sides by $\log_{10} 3$, we get:

$$\log_{10} x = 2 \log_{10} 3$$

Using the power rule, we can rewrite the equation as:

$$\log_{10} x = \log_{10} 3^2$$

Therefore, $x = 3^2 = 9$.

Conclusion

Q: What is the change of base formula?

A: The change of base formula is a fundamental concept in logarithmic equivalents. It states that:

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

where $a$, $b$, and $c$ are positive real numbers, and $c \neq 1$.

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

A: To apply the change of base formula, you need to choose a new base and rewrite the original logarithmic expression using that base. For example, if you want to convert a logarithmic expression from base 2 to base 10, you would use the change of base formula as follows:

$$\log_2 a = \frac{\log_{10} a}{\log_{10} 2}$$

Q: What are some common logarithmic equivalents?

A: Some common logarithmic equivalents include:

• $\log_2 a = \frac{\log_{10} a}{\log_{10} 2}$
• $\log_3 a = \frac{\log_{10} a}{\log_{10} 3}$
• $\log_e a = \frac{\ln a}{\ln e}$

Q: How do I simplify logarithmic expressions?

A: To simplify logarithmic expressions, you can use the properties of logarithms, such as the product rule, quotient rule, and power rule. For example, if you have the expression $\log (ab)$, you can simplify it using the product rule as follows:

$$\log (ab) = \log a + \log b$$

Q: Can I use logarithmic equivalents to solve logarithmic equations?

A: Yes, you can use logarithmic equivalents to solve logarithmic equations. By applying the change of base formula and simplifying the expression, you can solve logarithmic equations involving different bases.

Q: What are some real-world applications of logarithmic equivalents?

A: Logarithmic equivalents have numerous real-world applications, including:

• Finance: Logarithmic equivalents are used to calculate interest rates and investment returns.
• Science: Logarithmic equivalents are used to calculate pH levels and concentrations of solutions.
• Engineering: Logarithmic equivalents are used to calculate signal strengths and noise levels.

Q: Can I use logarithmic equivalents with different bases?

A: Yes, you can use logarithmic equivalents with different bases. The change of base formula allows you to convert logarithmic expressions from one base to another.

Q: How do I choose the right base for a logarithmic expression?

A: The choice of base depends on the specific problem and the units of measurement. For example, if you are working with pH levels, you would use base 10. If you are working with signal strengths, you would use base 2.

Q: Can I use logarithmic equivalents with negative numbers?

A: No, you cannot use logarithmic equivalents with negative numbers. Logarithmic functions are only defined for positive real numbers.

Q: Can I use logarithmic equivalents with complex numbers?

A: Yes, you can use logarithmic equivalents with complex numbers. However, the change of base formula may not be applicable in all cases.

Conclusion

In conclusion, logarithmic equivalents are a powerful tool for converting logarithmic expressions from one base to another. By applying the change of base formula and simplifying the expression, you can solve logarithmic equations and calculate logarithmic values with different bases.