Which Of The Following Is Equivalent To $\log_3 8$?A. $\log_8 3$ B. $2 \log_3 3$ C. $3 \log_3 2$ D. 2

Introduction

Logarithmic equations are a fundamental concept in mathematics, and they play a crucial role in various fields, including physics, engineering, and computer science. In this article, we will explore the concept of logarithmic equations and provide a step-by-step guide on how to solve them. We will also discuss the properties of logarithms and how to apply them to solve complex equations.

What are Logarithmic Equations?

A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. In other words, a logarithmic equation is an equation that involves a variable raised to a power, and the result is equal to a given value. For example, the equation $\log_3 8 = x$ is a logarithmic equation, where $x$ is the variable and $3$ is the base of the logarithm.

Properties of Logarithms

There are several properties of logarithms that we need to understand in order to solve logarithmic equations. These properties include:

• Product Rule: $\log_b (xy) = \log_b x + \log_b y$
• Quotient Rule: $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$
• Power Rule: $\log_b x^y = y \log_b x$
• Change of Base Rule: $\log_b x = \frac{\log_a x}{\log_a b}$

Solving Logarithmic Equations

Now that we have discussed the properties of logarithms, let's move on to solving logarithmic equations. There are several steps that we need to follow in order to solve a logarithmic equation:

1. Isolate the logarithm: The first step in solving a logarithmic equation is to isolate the logarithm. This means that we need to get the logarithm by itself on one side of the equation.
2. Apply the properties of logarithms: Once we have isolated the logarithm, we can apply the properties of logarithms to simplify the equation.
3. Solve for the variable: Finally, we can solve for the variable by applying the inverse operation of the logarithm.

Example 1: Solving a Logarithmic Equation

Let's consider the equation $\log_3 8 = x$. To solve this equation, we need to isolate the logarithm and apply the properties of logarithms.

• Isolate the logarithm: The first step is to isolate the logarithm by getting rid of the $x$ on the right-hand side of the equation. We can do this by applying the inverse operation of the logarithm, which is exponentiation.
• Apply the properties of logarithms: Once we have isolated the logarithm, we can apply the properties of logarithms to simplify the equation. In this case, we can use the power rule to rewrite the equation as $3^x = 8$.
• Solve for the variable: Finally, we can solve for the variable by applying the inverse operation of the logarithm, which is exponentiation. In this case, we can take the logarithm of both sides of the equation to get $x = \log_3 8$.

Which of the Following is Equivalent to $\log_3 8$?

Now that we have discussed how to solve logarithmic equations, let's move on to the main question of this article, which is to determine which of the following is equivalent to $\log_3 8$.

A. $\log_8 3$ B. $2 \log_3 3$ C. $3 \log_3 2$ D. 2

To determine which of the following is equivalent to $\log_3 8$, we need to apply the properties of logarithms and simplify each option.

• Option A: $\log_8 3$ is equivalent to $\frac{1}{\log_3 8}$, which is not equal to $\log_3 8$.
• Option B: $2 \log_3 3$ is equivalent to $\log_3 3^2$, which is equal to $\log_3 9$. This is not equal to $\log_3 8$.
• Option C: $3 \log_3 2$ is equivalent to $\log_3 2^3$, which is equal to $\log_3 8$. This is the correct answer.
• Option D: 2 is not equal to $\log_3 8$.

Conclusion

In conclusion, we have discussed the concept of logarithmic equations and provided a step-by-step guide on how to solve them. We have also discussed the properties of logarithms and how to apply them to solve complex equations. Finally, we have determined which of the following is equivalent to $\log_3 8$, and the correct answer is $3 \log_3 2$.