Which Of The Following Uses The Properties Of Logarithms To Correctly Rewrite $f(x)=\log _2\left(32 X^6\right)$?A. F ( X ) = 6 Log ⁡ 2 X + 5 F(x)=6 \log _2 X+5 F ( X ) = 6 Lo G 2 ​ X + 5 B. F ( X ) = 30 Log ⁡ 2 X F(x)=30 \log _2 X F ( X ) = 30 Lo G 2 ​ X C. F ( X ) = 6 Log ⁡ 2 X + 30 F(x)=6 \log _2 X+30 F ( X ) = 6 Lo G 2 ​ X + 30 D. $f(x)=\log _2 6

Introduction

Logarithms are a fundamental concept in mathematics, and understanding their properties is crucial for solving various mathematical problems. In this article, we will delve into the world of logarithms and explore how to correctly rewrite a given function using the properties of logarithms.

What are Logarithms?

A logarithm is the inverse operation of exponentiation. It is a mathematical function that takes a number as input and returns the power to which a base number must be raised to produce the input number. In other words, if $y = \log_b(x)$, then $b^y = x$. The base of a logarithm is a positive real number other than 1.

Properties of Logarithms

There are several properties of logarithms that are essential to understand when rewriting a function using logarithms. These properties include:

• Product Rule: $\log_b(xy) = \log_b(x) + \log_b(y)$
• Quotient Rule: $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$
• Power Rule: $\log_b(x^y) = y \log_b(x)$

Rewriting the Function

Now that we have a good understanding of the properties of logarithms, let's apply them to rewrite the given function:

$$f(x)=\log _2\left(32 x^6\right)$$

Using the Power Rule, we can rewrite the function as:

$$f(x)=\log _2\left(32 x^6\right) = \log _2(32) + \log _2(x^6)$$

Next, we can use the Product Rule to rewrite $\log _2(32)$ as:

$$\log _2(32) = \log _2(2^5) = 5 \log _2(2)$$

Since $\log _2(2) = 1$, we can simplify the expression to:

$$\log _2(32) = 5$$

Now, let's focus on the second term, $\log _2(x^6)$. Using the Power Rule again, we can rewrite it as:

$$\log _2(x^6) = 6 \log _2(x)$$

Therefore, the rewritten function is:

$$f(x) = 5 + 6 \log _2(x)$$

Comparing the Options

Now that we have rewritten the function using the properties of logarithms, let's compare it with the given options:

• A. $f(x)=6 \log _2 x+5$
• B. $f(x)=30 \log _2 x$
• C. $f(x)=6 \log _2 x+30$
• D. $f(x)=\log _2 6$

The correct option is A. $f(x)=6 \log _2 x+5$. This option correctly reflects the rewritten function we obtained using the properties of logarithms.

Conclusion

In this article, we explored the properties of logarithms and applied them to rewrite a given function. We used the Product Rule, Quotient Rule, and Power Rule to simplify the expression and obtain the correct rewritten function. By understanding the properties of logarithms, we can solve various mathematical problems and rewrite functions in a more concise and elegant way.

Final Answer

$$

Q&A: Properties of Logarithms

Q: What is the product rule of logarithms?

A: The product rule of logarithms states that $\log_b(xy) = \log_b(x) + \log_b(y)$. This means that the logarithm of a product is equal to the sum of the logarithms of the individual factors.

Q: How do I apply the product rule to a logarithmic expression?

A: To apply the product rule, simply break down the logarithmic expression into its individual factors and then add the logarithms of each factor. For example, $\log_2(4x) = \log_2(4) + \log_2(x)$.

Q: What is the quotient rule of logarithms?

A: The quotient rule of logarithms states that $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$. This means that the logarithm of a quotient is equal to the difference of the logarithms of the individual factors.

Q: How do I apply the quotient rule to a logarithmic expression?

A: To apply the quotient rule, simply break down the logarithmic expression into its individual factors and then subtract the logarithm of the divisor from the logarithm of the dividend. For example, $\log_2(\frac{4}{x}) = \log_2(4) - \log_2(x)$.

Q: What is the power rule of logarithms?

A: The power rule of logarithms states that $\log_b(x^y) = y \log_b(x)$. This means that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base.

Q: How do I apply the power rule to a logarithmic expression?

A: To apply the power rule, simply multiply the exponent by the logarithm of the base. For example, $\log_2(x^3) = 3 \log_2(x)$.

Q: Can I use the properties of logarithms to simplify a logarithmic expression?

A: Yes, you can use the properties of logarithms to simplify a logarithmic expression. By applying the product rule, quotient rule, and power rule, you can break down the expression into its individual factors and then simplify.

Q: How do I rewrite a logarithmic expression using the properties of logarithms?

A: To rewrite a logarithmic expression using the properties of logarithms, simply apply the product rule, quotient rule, and power rule to break down the expression into its individual factors. Then, simplify the expression by combining like terms.

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

A: Some common mistakes to avoid when working with logarithmic expressions include:

• Forgetting to apply the product rule when multiplying two logarithmic expressions
• Forgetting to apply the quotient rule when dividing two logarithmic expressions
• Forgetting to apply the power rule when raising a logarithmic expression to a power
• Not simplifying the expression by combining like terms

Q: How do I check my work when rewriting a logarithmic expression using the properties of logarithms?

A: To check your work, simply plug in a value for the variable and evaluate the expression. If the expression is true for all values of the variable, then your work is correct.

Conclusion

In this article, we explored the properties of logarithms and answered some common questions about logarithmic expressions. By understanding the product rule, quotient rule, and power rule, you can simplify and rewrite logarithmic expressions with ease. Remember to always check your work by plugging in a value for the variable and evaluating the expression.