If Log ⁡ 2 ( 3 X + 6 ) = 1 \log_2(3x + 6) = 1 Lo G 2 ​ ( 3 X + 6 ) = 1 , Then X = □ X = \square X = □ .You May Enter The Exact Value Or Round To 4 Decimal Places.

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Introduction

In this article, we will explore the concept of logarithms and how to solve equations involving logarithms. We will use the given equation $\log_2(3x + 6) = 1$ to find the value of $x$. This equation involves a logarithm with base 2, and we will use the properties of logarithms to solve for $x$.

Understanding Logarithms

A logarithm is the inverse operation of exponentiation. In other words, if $a^b = c$, then $\log_a(c) = b$. The logarithm of a number is the exponent to which the base must be raised to produce that number. For example, $\log_2(8) = 3$ because $2^3 = 8$.

Solving the Equation

To solve the equation $\log_2(3x + 6) = 1$, we can start by rewriting the equation in exponential form. Since the base of the logarithm is 2, we can rewrite the equation as:

$$2^1 = 3x + 6$$

Simplifying the Equation

Now we can simplify the equation by evaluating the left-hand side:

$$2 = 3x + 6$$

Isolating $x$

Next, we can isolate $x$ by subtracting 6 from both sides of the equation:

$$-4 = 3x$$

Solving for $x$

Finally, we can solve for $x$ by dividing both sides of the equation by 3:

$$x = -\frac{4}{3}$$

Conclusion

In this article, we used the properties of logarithms to solve the equation $\log_2(3x + 6) = 1$. We started by rewriting the equation in exponential form, then simplified the equation and isolated $x$. Finally, we solved for $x$ and found that $x = -\frac{4}{3}$.

Example Use Case

This equation can be used in a variety of real-world applications, such as:

• Finance: In finance, logarithmic equations can be used to model the growth of investments over time. For example, if an investment grows at a rate of 10% per year, the logarithmic equation can be used to calculate the future value of the investment.
• Science: In science, logarithmic equations can be used to model the behavior of physical systems. For example, the logarithmic equation can be used to model the growth of populations over time.
• Engineering: In engineering, logarithmic equations can be used to model the behavior of complex systems. For example, the logarithmic equation can be used to model the behavior of electronic circuits.

Tips and Tricks

Here are some tips and tricks for solving logarithmic equations:

• Use the properties of logarithms: The properties of logarithms can be used to simplify logarithmic equations and make them easier to solve.
• Rewrite the equation in exponential form: Rewriting the equation in exponential form can make it easier to solve and understand.
• Simplify the equation: Simplifying the equation can make it easier to solve and understand.
• Isolate $x$: Isolating $x$ can make it easier to solve for $x$.

Common Mistakes

Here are some common mistakes to avoid when solving logarithmic equations:

• Forgetting to rewrite the equation in exponential form: Forgetting to rewrite the equation in exponential form can make it difficult to solve and understand.
• Not simplifying the equation: Not simplifying the equation can make it difficult to solve and understand.
• Not isolating $x$: Not isolating $x$ can make it difficult to solve for $x$.

Conclusion

In conclusion, solving logarithmic equations can be a challenging but rewarding task. By using the properties of logarithms, rewriting the equation in exponential form, simplifying the equation, and isolating $x$, we can solve logarithmic equations and find the value of $x$. With practice and patience, anyone can become proficient in solving logarithmic equations.

Final Answer

The final answer is: $\boxed{-\frac{4}{3}}$

Introduction

In this article, we will answer some common questions about logarithmic equations. We will cover topics such as how to solve logarithmic equations, how to use the properties of logarithms, and how to apply logarithmic equations to real-world problems.

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithm. A logarithm is the inverse operation of exponentiation. In other words, if $a^b = c$, then $\log_a(c) = b$. Logarithmic equations can be used to model a wide range of real-world problems, from finance to science to engineering.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you can start by rewriting the equation in exponential form. This involves using the property of logarithms that states $\log_a(b) = c$ is equivalent to $a^c = b$. Once you have rewritten the equation in exponential form, you can simplify the equation and isolate the variable.

Q: What are some common properties of logarithms?

A: There are several common properties of logarithms that can be used to simplify logarithmic equations. These include:

• Product property: $\log_a(b \cdot c) = \log_a(b) + \log_a(c)$
• Quotient property: $\log_a(\frac{b}{c}) = \log_a(b) - \log_a(c)$
• Power property: $\log_a(b^c) = c \cdot \log_a(b)$

Q: How do I use the properties of logarithms to simplify a logarithmic equation?

A: To use the properties of logarithms to simplify a logarithmic equation, you can start by identifying the properties that can be applied to the equation. For example, if the equation involves a product or quotient, you can use the product or quotient property to simplify the equation. If the equation involves a power, you can use the power property to simplify the equation.

Q: Can logarithmic equations be used to model real-world problems?

A: Yes, logarithmic equations can be used to model a wide range of real-world problems. For example, logarithmic equations can be used to model the growth of populations, the behavior of physical systems, and the growth of investments.

Q: What are some examples of real-world problems that can be modeled using logarithmic equations?

A: Some examples of real-world problems that can be modeled using logarithmic equations include:

• Finance: Logarithmic equations can be used to model the growth of investments over time.
• Science: Logarithmic equations can be used to model the behavior of physical systems, such as the growth of populations or the behavior of chemical reactions.
• Engineering: Logarithmic equations can be used to model the behavior of complex systems, such as electronic circuits or mechanical systems.

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

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

• Forgetting to rewrite the equation in exponential form: Forgetting to rewrite the equation in exponential form can make it difficult to solve and understand.
• Not simplifying the equation: Not simplifying the equation can make it difficult to solve and understand.
• Not isolating the variable: Not isolating the variable can make it difficult to solve for the variable.

Q: How can I practice solving logarithmic equations?

A: There are several ways to practice solving logarithmic equations, including:

• Working through practice problems: Working through practice problems can help you develop your skills and build your confidence.
• Using online resources: There are many online resources available that can help you practice solving logarithmic equations, including video tutorials and interactive exercises.
• Seeking help from a tutor or teacher: If you are struggling to understand logarithmic equations, seeking help from a tutor or teacher can be a great way to get the help you need.

Conclusion

In conclusion, logarithmic equations can be a powerful tool for modeling real-world problems. By understanding the properties of logarithms and how to apply them to solve logarithmic equations, you can develop the skills you need to tackle a wide range of problems. With practice and patience, anyone can become proficient in solving logarithmic equations.