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

Mar 8, 2025 by ADMIN 247 views

$Which Statement is True for $\log _3(x+1)=2$?$

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 $3^2 = x+1$ .

Exponential Form

To solve the equation, we can rewrite it in exponential form. The exponential form of a logarithmic equation is obtained by raising the base to the power of the logarithm. In this case, we have:

\log _3(x+1)=2 \Rightarrow 3^2 = x+1

Evaluating the Options

Now that we have rewritten the equation in exponential form, we can evaluate the options provided.

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

This option states that $x+1=3^2$ . Since we have already rewritten the equation in exponential form 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 in exponential form 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 in exponential form 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 in exponential form 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 statement is:

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

This is because we have rewritten the equation in exponential form as $3^2 = x+1$ , which is equivalent to $x+1=3^2$ .

Final Answer

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

Additional Information

The base-3 logarithm is a logarithm with respect to the base 3.
The exponential form of a logarithmic equation is obtained by raising the base to the power of the logarithm.
The equation $\log _3(x+1)=2$ can be rewritten in exponential form as $3^2 = x+1$ .
The correct statement is $x+1=3^2$ .

References

[1] "Logarithmic Equations" by Math Open Reference
[2] "Exponential Form" by Khan Academy
[3] "Base-3 Logarithm" by Wolfram MathWorld

Understanding Logarithmic Equations

Logarithmic equations are equations that involve a logarithm. They can be solved using various methods, including rewriting the equation in exponential form and using properties of logarithms.

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithm. It can be written in the form $\log _b(x) = y$ , where $b$ is the base of the logarithm, $x$ is the argument of the logarithm, and $y$ is the result of the logarithm.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you can rewrite it in exponential form and then solve for the variable. For example, the equation $\log _3(x+1)=2$ can be rewritten in exponential form as $3^2 = x+1$ .

Q: What is the exponential form of a logarithmic equation?

A: The exponential form of a logarithmic equation is obtained by raising the base to the power of the logarithm. For example, the equation $\log _3(x+1)=2$ can be rewritten in exponential form as $3^2 = x+1$ .

Q: How do I evaluate the options in a logarithmic equation?

A: To evaluate the options in a logarithmic equation, you can substitute each option into the equation and check if it is true. For example, in the equation $\log _3(x+1)=2$ , you can substitute each option into the equation and check if it is true.

Q: What is the base-3 logarithm?

A: The base-3 logarithm is a logarithm with respect to the base 3. It is denoted by $\log _3(x)$ and is equal to the power to which 3 must be raised to produce the number $x$ .

Q: How do I use properties of logarithms to solve an equation?

A: To use properties of logarithms to solve an equation, you can use the following properties:

$\log _b(xy) = \log _b(x) + \log _b(y)$
$\log _b(\frac{x}{y}) = \log _b(x) - \log _b(y)$
$\log _b(x^y) = y\log _b(x)$

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

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

Not rewriting the equation in exponential form
Not using properties of logarithms
Not checking the options in the equation

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

A: To check your answer in a logarithmic equation, you can substitute the answer into the equation and check if it is true. For example, if you solve the equation $\log _3(x+1)=2$ and get $x+1=9$ , you can substitute $x+1=9$ into the equation and check if it is true.

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 calculate the pH of a solution and the concentration of a substance.
Engineering: Logarithmic equations are used to calculate the power of a signal and the frequency of a wave.

Q: How do I use logarithmic equations in real-world applications?

A: To use logarithmic equations in real-world applications, you can use the following steps:

Identify the problem and the variables involved
Write the equation in logarithmic form
Solve the equation using properties of logarithms
Check the answer and interpret the results

Q: What are some common types of logarithmic equations?

A: Some common types of logarithmic equations include:

Base-10 logarithmic equations
Base-2 logarithmic equations
Base-3 logarithmic equations
Logarithmic equations with a variable base

Q: How do I solve logarithmic equations with a variable base?

A: To solve logarithmic equations with a variable base, you can use the following steps:

Identify the base and the argument of the logarithm
Rewrite the equation in exponential form
Solve for the variable using properties of logarithms
Check the answer and interpret the results

Q: What are some common mistakes to avoid when solving logarithmic equations with a variable base?

A: Some common mistakes to avoid when solving logarithmic equations with a variable base include:

Not identifying the base and the argument of the logarithm
Not rewriting the equation in exponential form
Not using properties of logarithms

Q: How do I check my answer in a logarithmic equation with a variable base?

A: To check your answer in a logarithmic equation with a variable base, you can substitute the answer into the equation and check if it is true. For example, if you solve the equation $\log _x(x+1)=2$ and get $x=3$ , you can substitute $x=3$ into the equation and check if it is true.

Q: What are some real-world applications of logarithmic equations with a variable base?

A: Logarithmic equations with a variable base have many real-world applications, including:

Finance: Logarithmic equations with a variable base are used to calculate interest rates and investment returns.
Science: Logarithmic equations with a variable base are used to calculate the pH of a solution and the concentration of a substance.
Engineering: Logarithmic equations with a variable base are used to calculate the power of a signal and the frequency of a wave.

Q: How do I use logarithmic equations with a variable base in real-world applications?

A: To use logarithmic equations with a variable base in real-world applications, you can use the following steps:

Identify the problem and the variables involved
Write the equation in logarithmic form
Solve the equation using properties of logarithms
Check the answer and interpret the results

Q: What are some common types of logarithmic equations with a variable base?

A: Some common types of logarithmic equations with a variable base include:

Logarithmic equations with a variable base and a constant argument
Logarithmic equations with a constant base and a variable argument
Logarithmic equations with a variable base and a variable argument

Q: How do I solve logarithmic equations with a variable base and a constant argument?

A: To solve logarithmic equations with a variable base and a constant argument, you can use the following steps:

Identify the base and the argument of the logarithm
Rewrite the equation in exponential form
Solve for the variable using properties of logarithms
Check the answer and interpret the results

Q: What are some common mistakes to avoid when solving logarithmic equations with a variable base and a constant argument?

A: Some common mistakes to avoid when solving logarithmic equations with a variable base and a constant argument include:

Not identifying the base and the argument of the logarithm
Not rewriting the equation in exponential form
Not using properties of logarithms

Q: How do I check my answer in a logarithmic equation with a variable base and a constant argument?

A: To check your answer in a logarithmic equation with a variable base and a constant argument, you can substitute the answer into the equation and check if it is true. For example, if you solve the equation $\log _x(2)=1$ and get $x=2$ , you can substitute $x=2$ into the equation and check if it is true.

Q: What are some real-world applications of logarithmic equations with a variable base and a constant argument?

A: Logarithmic equations with a variable base and a constant argument have many real-world applications, including:

Finance: Logarithmic equations with a variable base and a constant argument are used to calculate interest rates and investment returns.
Science: Logarithmic equations with a variable base and a constant argument are used to calculate the pH of a solution and the concentration of a substance.
Engineering: Logarithmic equations with a variable base and a constant argument are used to calculate the power of a signal and the frequency of a wave.

Q: How do I use logarithmic equations with a variable base and a constant argument in real-world applications?

A: To use logarithmic equations with a variable base and a constant argument in real-world applications, you can use the following steps:

Identify the problem and the variables involved
Write the equation in logarithmic form
Solve the equation using properties of logarithms
Check the answer and interpret the results

Q: What are some common types of logarithmic equations with a constant base and a variable argument?

A: Some common types of logarithmic equations with a constant base and a variable argument include:

Logarithmic equations with a base of 10 and a variable argument
Logarithmic equations with a base of 2 and a variable argument
Logarithmic equations with a base of 3 and a variable argument

Q: How do I solve logarithmic equations with a constant base and a variable argument?

A: To solve logarithmic equations with a constant base and a variable argument, you can use the following steps:

Identify the base and the argument of the logarithm
Rewrite the equation in exponential form
Solve for the variable using properties of logarithms
Check the answer and interpret the results

Q: What are some common mistakes to avoid when solving logarithmic equations with a constant base and a variable argument?

A: Some common mistakes