What Are The Solutions To The Equation:$\log_3 X + \log_3 (x^2 + 2) = 1 + 2\log_3 X$?A. X = -2 B. X = -1 C. X = 1 D. X = 2 E. There Is No True Solution.

Feb 26, 2025 by ADMIN 157 views

$What are the solutions to the equation: $\log_3 x + \log_3 (x^2 + 2) = 1 + 2\log_3 x$?$

Introduction

The given equation involves logarithmic functions and requires a step-by-step approach to solve it. The equation is $\log_3 x + \log_3 (x^2 + 2) = 1 + 2\log_3 x$ . Our goal is to find the value of $x$ that satisfies this equation. We will use various properties of logarithms to simplify the equation and solve for $x$ .

Step 1: Simplify the Equation Using Logarithmic Properties

We can start by using the property of logarithms that states $\log_a b + \log_a c = \log_a (bc)$ . Applying this property to the given equation, we get:

$\log_3 (x(x^2 + 2)) = 1 + 2\log_3 x$

Step 2: Use the Power Rule of Logarithms

Next, we can use the power rule of logarithms, which states $\log_a b^c = c\log_a b$ . Applying this rule to the equation, we get:

$\log_3 (x(x^2 + 2)) = \log_3 (3^1) + 2\log_3 x$

Step 3: Simplify the Right-Hand Side of the Equation

We can simplify the right-hand side of the equation by evaluating $\log_3 (3^1)$ , which is equal to 1. The equation becomes:

$\log_3 (x(x^2 + 2)) = 1 + 2\log_3 x$

Step 4: Use the One-to-One Property of Logarithms

Since the logarithmic function is one-to-one, we can equate the arguments of the logarithmic functions on both sides of the equation. This gives us:

$x(x^2 + 2) = 3^{1 + 2\log_3 x}$

Step 5: Simplify the Right-Hand Side of the Equation

We can simplify the right-hand side of the equation by evaluating $3^{1 + 2\log_3 x}$ . Using the property of exponents that states $a^{b+c} = a^b \cdot a^c$ , we get:

$x(x^2 + 2) = 3 \cdot 3^{2\log_3 x}$

Step 6: Simplify the Right-Hand Side of the Equation

We can simplify the right-hand side of the equation by evaluating $3^{2\log_3 x}$ . Using the property of logarithms that states $\log_a b^c = c\log_a b$ , we get:

$x(x^2 + 2) = 3 \cdot x^2$

Step 7: Simplify the Equation

We can simplify the equation by dividing both sides by $x$ . This gives us:

$x^2 + 2 = 3x$

Step 8: Rearrange the Equation

We can rearrange the equation to get a quadratic equation in $x$ . This gives us:

$x^2 - 3x + 2 = 0$

Step 9: Factor the Quadratic Equation

We can factor the quadratic equation to get:

$(x - 2)(x - 1) = 0$

Step 10: Solve for x

We can solve for $x$ by setting each factor equal to zero. This gives us:

$x - 2 = 0 \Rightarrow x = 2$

$x - 1 = 0 \Rightarrow x = 1$

Conclusion

The solutions to the equation $\log_3 x + \log_3 (x^2 + 2) = 1 + 2\log_3 x$ are $x = 1$ and $x = 2$ . Therefore, the correct answer is:

The final answer is: D. x = 2 and C. x = 1

Introduction

Q&A

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

A: The first step in solving the equation is to simplify the equation using logarithmic properties. We can use the property of logarithms that states $\log_a b + \log_a c = \log_a (bc)$ . Applying this property to the given equation, we get:

$\log_3 (x(x^2 + 2)) = 1 + 2\log_3 x$

Q: What is the next step in solving the equation?

A: The next step in solving the equation is to use the power rule of logarithms. The power rule of logarithms states that $\log_a b^c = c\log_a b$ . Applying this rule to the equation, we get:

$\log_3 (x(x^2 + 2)) = \log_3 (3^1) + 2\log_3 x$

Q: How do we simplify the right-hand side of the equation?

A: We can simplify the right-hand side of the equation by evaluating $\log_3 (3^1)$ , which is equal to 1. The equation becomes:

$\log_3 (x(x^2 + 2)) = 1 + 2\log_3 x$

Q: What is the one-to-one property of logarithms?

A: The one-to-one property of logarithms states that if $\log_a x = \log_a y$ , then $x = y$ . We can use this property to equate the arguments of the logarithmic functions on both sides of the equation. This gives us:

$x(x^2 + 2) = 3^{1 + 2\log_3 x}$

Q: How do we simplify the right-hand side of the equation?

A: We can simplify the right-hand side of the equation by evaluating $3^{1 + 2\log_3 x}$ . Using the property of exponents that states $a^{b+c} = a^b \cdot a^c$ , we get:

$x(x^2 + 2) = 3 \cdot 3^{2\log_3 x}$

Q: How do we simplify the right-hand side of the equation?

A: We can simplify the right-hand side of the equation by evaluating $3^{2\log_3 x}$ . Using the property of logarithms that states $\log_a b^c = c\log_a b$ , we get:

$x(x^2 + 2) = 3 \cdot x^2$

Q: How do we simplify the equation?

A: We can simplify the equation by dividing both sides by $x$ . This gives us:

$x^2 + 2 = 3x$

Q: How do we rearrange the equation?

A: We can rearrange the equation to get a quadratic equation in $x$ . This gives us:

$x^2 - 3x + 2 = 0$

Q: How do we factor the quadratic equation?

A: We can factor the quadratic equation to get:

$(x - 2)(x - 1) = 0$

Q: How do we solve for x?

A: We can solve for $x$ by setting each factor equal to zero. This gives us:

$x - 2 = 0 \Rightarrow x = 2$

$x - 1 = 0 \Rightarrow x = 1$

Conclusion

The solutions to the equation $\log_3 x + \log_3 (x^2 + 2) = 1 + 2\log_3 x$ are $x = 1$ and $x = 2$ . Therefore, the correct answer is:

The final answer is: D. x = 2 and C. x = 1

Frequently Asked Questions

Q: What is the main concept used to solve the equation?

A: The main concept used to solve the equation is the properties of logarithms.

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

A: The first step in solving the equation is to simplify the equation using logarithmic properties.

Q: How do we simplify the right-hand side of the equation?

A: We can simplify the right-hand side of the equation by evaluating the logarithmic expressions.

Q: What is the one-to-one property of logarithms?

A: The one-to-one property of logarithms states that if $\log_a x = \log_a y$ , then $x = y$ .

Q: How do we solve for x?

A: We can solve for $x$ by setting each factor equal to zero.

Final Answer

The final answer is: D. x = 2 and C. x = 1