What Is The Solution Of $\log_2(3x-7)=3$?A. $\frac{1}{3}$ B. 4 C. 5 D. $\frac{16}{3}$

Introduction

Logarithmic equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will focus on solving the equation $\log_2(3x-7)=3$. This equation involves a logarithm with base 2, and our goal is to isolate the variable $x$.

Understanding Logarithms

Before we dive into solving the equation, let's take a moment to understand what logarithms are. A logarithm is the inverse operation of exponentiation. In other words, if $a^b = c$, then $\log_a(c) = b$. The logarithm with base 2, denoted as $\log_2(x)$, is the exponent to which 2 must be raised to produce the number $x$.

Solving the Equation

Now that we have a basic understanding of logarithms, let's tackle the equation $\log_2(3x-7)=3$. To solve this equation, we need to get rid of the logarithm. We can do this by using the definition of logarithms.

Step 1: Exponentiate Both Sides

The first step in solving the equation is to exponentiate both sides. This means that we will raise 2 to the power of both sides of the equation. This will eliminate the logarithm.

$$\log_2(3x-7)=3 \Rightarrow 2^{\log_2(3x-7)}=2^3$$

Step 2: Simplify the Equation

Now that we have exponentiated both sides, we can simplify the equation. The left-hand side of the equation can be simplified using the property of logarithms that states $a^{\log_a(x)}=x$.

$$2^{\log_2(3x-7)}=2^3 \Rightarrow 3x-7=8$$

Step 3: Isolate the Variable

The final step in solving the equation is to isolate the variable $x$. We can do this by adding 7 to both sides of the equation and then dividing both sides by 3.

$$3x-7=8 \Rightarrow 3x=15 \Rightarrow x=5$$

Conclusion

In this article, we solved the equation $\log_2(3x-7)=3$ using a step-by-step approach. We first exponentiated both sides of the equation to eliminate the logarithm, then simplified the equation, and finally isolated the variable $x$. The solution to the equation is $x=5$.

Answer

The correct answer is C. 5.

Additional Tips and Tricks

• When solving logarithmic equations, it's essential to remember the definition of logarithms and use it to eliminate the logarithm.
• Exponentiating both sides of the equation is a common technique used to solve logarithmic equations.
• Simplifying the equation after exponentiating both sides can help make the equation easier to solve.
• Isolating the variable is the final step in solving the equation, and it's essential to do it carefully to avoid making mistakes.

Common Mistakes to Avoid

• Not using the definition of logarithms to eliminate the logarithm.
• Not exponentiating both sides of the equation.
• Not simplifying the equation after exponentiating both sides.
• Not isolating the variable carefully.

Real-World Applications

Logarithmic equations have many real-world applications, including:

• Finance: Logarithmic equations are used to calculate interest rates and investment returns.
• Science: Logarithmic equations are used to model population growth and decay.
• Engineering: Logarithmic equations are used to design electronic circuits and communication systems.

Conclusion

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Q: What is the definition of a logarithm?

A: A logarithm is the inverse operation of exponentiation. In other words, if $a^b = c$, then $\log_a(c) = b$. The logarithm with base 2, denoted as $\log_2(x)$, is the exponent to which 2 must be raised to produce the number $x$.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to get rid of the logarithm. You can do this by using the definition of logarithms and exponentiating both sides of the equation.

Q: What is the first step in solving a logarithmic equation?

A: The first step in solving a logarithmic equation is to exponentiate both sides of the equation. This means that you will raise the base of the logarithm to the power of both sides of the equation.

Q: How do I simplify the equation after exponentiating both sides?

A: After exponentiating both sides of the equation, you can simplify the equation by using the properties of logarithms. For example, if you have $\log_a(x) = b$, then $a^b = x$.

Q: How do I isolate the variable in a logarithmic equation?

A: To isolate the variable in a logarithmic equation, you need to get rid of the logarithm and the constant term. You can do this by adding or subtracting the constant term from both sides of the equation and then dividing both sides by the coefficient of the variable.

Q: What are some common mistakes to avoid when solving logarithmic equations?

A: Some common mistakes to avoid when solving logarithmic equations include:

• Not using the definition of logarithms to eliminate the logarithm.
• Not exponentiating both sides of the equation.
• Not simplifying the equation after exponentiating both sides.
• Not isolating the variable carefully.

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

A: Logarithmic equations have many real-world applications, including:

• Finance: Logarithmic equations are used to calculate interest rates and investment returns.
• Science: Logarithmic equations are used to model population growth and decay.
• Engineering: Logarithmic equations are used to design electronic circuits and communication systems.

Q: How do I check my answer when solving a logarithmic equation?

A: To check your answer when solving a logarithmic equation, you need to plug your solution back into the original equation and make sure that it is true. If your solution satisfies the original equation, then it is correct.

Q: What are some tips for solving logarithmic equations?

A: Some tips for solving logarithmic equations include:

• Using the definition of logarithms to eliminate the logarithm.
• Exponentiating both sides of the equation.
• Simplifying the equation after exponentiating both sides.
• Isolating the variable carefully.
• Checking your answer by plugging it back into the original equation.

Q: Can you give an example of a logarithmic equation?

A: Yes, here is an example of a logarithmic equation:

$\log_2(3x-7)=3$

This equation can be solved using the steps outlined above.

Q: Can you solve the equation $\log_2(3x-7)=3$?

A: Yes, the equation $\log_2(3x-7)=3$ can be solved as follows:

$\log_2(3x-7)=3 \Rightarrow 2^{\log_2(3x-7)}=2^3 \Rightarrow 3x-7=8 \Rightarrow 3x=15 \Rightarrow x=5$

Therefore, the solution to the equation $\log_2(3x-7)=3$ is $x=5$.