Which Statement Is True For Log ⁡ 3 ( X + 1 ) = 2 \log _3(x+1)=2 Lo G 3 ​ ( X + 1 ) = 2 ?A. X + 1 = 3 2 X+1=3^2 X + 1 = 3 2 B. X + 1 = 2 3 X+1=2^3 X + 1 = 2 3 C. 2 ( X + 1 ) = 3 2(x+1)=3 2 ( X + 1 ) = 3 D. 3 ( X + 1 ) = 2 3(x+1)=2 3 ( X + 1 ) = 2

Understanding the Problem

The problem involves solving a logarithmic equation, specifically a base-3 logarithm. We are given the equation $\log _3(x+1)=2$ and need to determine which of the provided statements is true.

Logarithmic Equations

A logarithmic equation is an equation that involves a logarithm. In this case, we have a base-3 logarithm, which means that the logarithm is with respect to the base 3. The equation $\log _3(x+1)=2$ can be rewritten in exponential form as $3^2=x+1$.

Rewriting the Equation

To solve the equation, we can rewrite it in exponential form. Since the base of the logarithm is 3, we can rewrite the equation as $3^2=x+1$. This is because the logarithm and exponential functions are inverses of each other.

Evaluating the Options

Now that we have rewritten the equation, we can evaluate the options to determine which one is true.

Option A: $x+1=3^2$

This option states that $x+1=3^2$. Since we have already rewritten the equation as $3^2=x+1$, this option is true.

Option B: $x+1=2^3$

This option states that $x+1=2^3$. However, we have already rewritten the equation as $3^2=x+1$, which is not equal to $2^3$. Therefore, this option is false.

Option C: $2(x+1)=3$

This option states that $2(x+1)=3$. However, we have already rewritten the equation as $3^2=x+1$, which is not equal to $2(x+1)=3$. Therefore, this option is false.

Option D: $3(x+1)=2$

This option states that $3(x+1)=2$. However, we have already rewritten the equation as $3^2=x+1$, which is not equal to $3(x+1)=2$. Therefore, this option is false.

Conclusion

Based on the analysis, the correct answer is option A: $x+1=3^2$. This is because we have rewritten the equation as $3^2=x+1$, which is equivalent to option A.

Additional Information

It's worth noting that the other options are not true because they do not match the rewritten equation. Option B is incorrect because $2^3$ is not equal to $3^2$. Option C is incorrect because $2(x+1)$ is not equal to $3$. Option D is incorrect because $3(x+1)$ is not equal to $2$.

Final Answer

The final answer is option A: $x+1=3^2$.

Understanding Logarithmic Equations

A logarithmic equation is an equation that involves a logarithm. In this case, we have a base-3 logarithm, which means that the logarithm is with respect to the base 3. The equation $\log _3(x+1)=2$ can be rewritten in exponential form as $3^2=x+1$.

Q&A

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithm. It is a mathematical expression that represents the power to which a base number must be raised to obtain a given value.

Q: How do I rewrite a logarithmic equation in exponential form?

A: To rewrite a logarithmic equation in exponential form, you need to use the fact that the logarithm and exponential functions are inverses of each other. For example, if you have the equation $\log _3(x+1)=2$, you can rewrite it in exponential form as $3^2=x+1$.

Q: What is the base of the logarithm in the equation $\log _3(x+1)=2$?

A: The base of the logarithm in the equation $\log _3(x+1)=2$ is 3.

Q: How do I evaluate the options in the equation $\log _3(x+1)=2$?

A: To evaluate the options, you need to rewrite the equation in exponential form and then compare it with each option. For example, if you have the equation $\log _3(x+1)=2$, you can rewrite it in exponential form as $3^2=x+1$. Then, you can compare it with each option to determine which one is true.

Q: What is the correct answer for the equation $\log _3(x+1)=2$?

A: The correct answer for the equation $\log _3(x+1)=2$ is option A: $x+1=3^2$.

Q: Why is option B incorrect?

A: Option B is incorrect because $2^3$ is not equal to $3^2$.

Q: Why is option C incorrect?

A: Option C is incorrect because $2(x+1)$ is not equal to $3$.

Q: Why is option D incorrect?

A: Option D is incorrect because $3(x+1)$ is not equal to $2$.

Conclusion

In conclusion, logarithmic equations are mathematical expressions that represent the power to which a base number must be raised to obtain a given value. To rewrite a logarithmic equation in exponential form, you need to use the fact that the logarithm and exponential functions are inverses of each other. The base of the logarithm in the equation $\log _3(x+1)=2$ is 3. To evaluate the options, you need to rewrite the equation in exponential form and then compare it with each option. The correct answer for the equation $\log _3(x+1)=2$ is option A: $x+1=3^2$.

Additional Information

It's worth noting that logarithmic equations can be used to solve a wide range of mathematical problems, including problems involving exponential growth and decay. They are also used in many real-world applications, such as finance, science, and engineering.

Final Answer

The final answer is option A: $x+1=3^2$.