Select The Correct Answer From The Drop-down Menu.Given:${ \log_a 2 = 0.3812, \quad \log_a 3 = 0.6013, \quad \text{and} \quad \log_a 5 = 0.9004 }$The Value Of { \log_a (30a)^3$}$ Is { \square$}$.Options:A. 3.169 B.

Introduction

Logarithmic equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of logarithms. In this article, we will explore how to solve logarithmic equations using the given values of $\log_a 2$, $\log_a 3$, and $\log_a 5$. We will also learn how to evaluate the value of $\log_a (30a)^3$ using the given values.

Understanding Logarithmic Properties

Before we dive into solving the equation, it's essential to understand the properties of logarithms. The logarithm of a number $x$ with base $a$ is denoted by $\log_a x$. The logarithmic function has several properties that we will use to solve the equation:

• Product Property: $\log_a (xy) = \log_a x + \log_a y$
• Quotient Property: $\log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y$
• Power Property: $\log_a x^y = y \log_a x$

Evaluating the Value of $\log_a (30a)^3$

To evaluate the value of $\log_a (30a)^3$, we can use the power property of logarithms. The power property states that $\log_a x^y = y \log_a x$. In this case, we have:

$$\log_a (30a)^3 = 3 \log_a (30a)$$

Now, we can use the product property to expand the expression:

$$3 \log_a (30a) = 3 (\log_a 30 + \log_a a)$$

Using the product property again, we can expand the expression further:

$$3 (\log_a 30 + \log_a a) = 3 (\log_a (30a))$$

Now, we can use the given values of $\log_a 2$, $\log_a 3$, and $\log_a 5$ to evaluate the expression:

$$3 (\log_a (30a)) = 3 (\log_a (2 \cdot 3 \cdot 5 \cdot a))$$

Using the product property, we can expand the expression:

$$3 (\log_a (2 \cdot 3 \cdot 5 \cdot a)) = 3 (\log_a 2 + \log_a 3 + \log_a 5 + \log_a a)$$

Now, we can substitute the given values of $\log_a 2$, $\log_a 3$, and $\log_a 5$:

$$3 (\log_a 2 + \log_a 3 + \log_a 5 + \log_a a) = 3 (0.3812 + 0.6013 + 0.9004 + 1)$$

Evaluating the expression, we get:

$$3 (0.3812 + 0.6013 + 0.9004 + 1) = 3 (2.883)$$

Simplifying the expression, we get:

$$3 (2.883) = 8.649$$

Therefore, the value of $\log_a (30a)^3$ is $\boxed{8.649}$.

Conclusion

In this article, we learned how to solve logarithmic equations using the given values of $\log_a 2$, $\log_a 3$, and $\log_a 5$. We also learned how to evaluate the value of $\log_a (30a)^3$ using the power property of logarithms. By understanding the properties of logarithms and applying them to the given equation, we were able to find the value of $\log_a (30a)^3$. This problem demonstrates the importance of logarithmic properties in solving complex equations.

Final Answer

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Introduction

Logarithmic equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of logarithms. In this article, we will explore some common questions and answers related to logarithmic equations.

Q: What is the product property of logarithms?

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

Q: How do I use the product property to solve a logarithmic equation?

A: To use the product property to solve a logarithmic equation, you need to identify the product inside the logarithm and break it down into its individual factors. Then, you can use the product property to rewrite the equation as the sum of the logarithms of the individual factors.

Q: What is the quotient property of logarithms?

A: The quotient property of logarithms states that $\log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y$. This means that the logarithm of a quotient is equal to the difference of the logarithms of the individual numbers.

Q: How do I use the quotient property to solve a logarithmic equation?

A: To use the quotient property to solve a logarithmic equation, you need to identify the quotient inside the logarithm and break it down into its individual factors. Then, you can use the quotient property to rewrite the equation as the difference of the logarithms of the individual factors.

Q: What is the power property of logarithms?

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

Q: How do I use the power property to solve a logarithmic equation?

A: To use the power property to solve a logarithmic equation, you need to identify the power inside the logarithm and break it down into its individual factors. Then, you can use the power property to rewrite the equation as the exponent multiplied by the logarithm of the base.

Q: How do I evaluate the value of $\log_a (30a)^3$?

A: To evaluate the value of $\log_a (30a)^3$, you can use the power property of logarithms. The power property states that $\log_a x^y = y \log_a x$. In this case, we have:

$$\log_a (30a)^3 = 3 \log_a (30a)$$

Now, you can use the product property to expand the expression:

$$3 \log_a (30a) = 3 (\log_a 30 + \log_a a)$$

Using the product property again, you can expand the expression further:

$$3 (\log_a 30 + \log_a a) = 3 (\log_a (30a))$$

Now, you can use the given values of $\log_a 2$, $\log_a 3$, and $\log_a 5$ to evaluate the expression:

$$3 (\log_a (30a)) = 3 (\log_a (2 \cdot 3 \cdot 5 \cdot a))$$

Using the product property, you can expand the expression:

$$3 (\log_a (2 \cdot 3 \cdot 5 \cdot a)) = 3 (\log_a 2 + \log_a 3 + \log_a 5 + \log_a a)$$

Now, you can substitute the given values of $\log_a 2$, $\log_a 3$, and $\log_a 5$:

$$3 (\log_a 2 + \log_a 3 + \log_a 5 + \log_a a) = 3 (0.3812 + 0.6013 + 0.9004 + 1)$$

Evaluating the expression, you get:

$$3 (0.3812 + 0.6013 + 0.9004 + 1) = 3 (2.883)$$

Simplifying the expression, you get:

$$3 (2.883) = 8.649$$

Therefore, the value of $\log_a (30a)^3$ is $\boxed{8.649}$.

Conclusion

In this article, we explored some common questions and answers related to logarithmic equations. We learned how to use the product property, quotient property, and power property to solve logarithmic equations. We also learned how to evaluate the value of $\log_a (30a)^3$ using the power property of logarithms. By understanding the properties of logarithms and applying them to the given equation, we were able to find the value of $\log_a (30a)^3$. This problem demonstrates the importance of logarithmic properties in solving complex equations.

Final Answer

The final answer is $\boxed{8.649}$.