Steps To Solve The Equation Log ⁡ 3 ( X + 2 ) = Log ⁡ 3 ( 2 X 2 − 1 \log_3(x+2)=\log_3(2x^2-1 Lo G 3 ​ ( X + 2 ) = Lo G 3 ​ ( 2 X 2 − 1 ]:1. Set The Arguments Of The Logarithms Equal To Each Other: ${ X+2 = 2x^2 - 1 }$2. Rearrange The Equation: ${ 0 = 2x^2 - X - 3 }$3. Factor The Quadratic

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Introduction

In this article, we will explore the steps to solve the equation $\log_3(x+2)=\log_3(2x^2-1)$. This equation involves logarithmic functions, and we will use properties of logarithms to simplify and solve it. We will start by setting the arguments of the logarithms equal to each other, then rearrange the equation to form a quadratic equation, and finally factor the quadratic to find the solutions.

Step 1: Set the Arguments of the Logarithms Equal to Each Other

The first step in solving the equation is to set the arguments of the logarithms equal to each other. This means that we equate the expressions inside the logarithms:

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

Since the logarithms have the same base, we can drop the logarithms and equate the expressions inside:

$$x+2=2x^2-1$$

This is the first step in solving the equation, and we have simplified it to a quadratic equation.

Step 2: Rearrange the Equation

The next step is to rearrange the equation to form a standard quadratic equation. We can do this by moving all the terms to one side of the equation:

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

This is a quadratic equation in the form $ax^2+bx+c=0$, where $a=2$, $b=-1$, and $c=-3$.

Step 3: Factor the Quadratic

The final step is to factor the quadratic equation. We can do this by finding two numbers whose product is $ac$ and whose sum is $b$. In this case, we need to find two numbers whose product is $2(-3)=-6$ and whose sum is $-1$.

After some trial and error, we find that the numbers are $-3$ and $2$, since $(-3)(2)=-6$ and $(-3)+2=-1$. Therefore, we can factor the quadratic as:

$$0=(x-3)(2x+1)$$

This is the factored form of the quadratic equation.

Solving the Equation

To solve the equation, we need to find the values of $x$ that make the factored expression equal to zero. We can do this by setting each factor equal to zero and solving for $x$:

$$x-3=0 \Rightarrow x=3$$

$$2x+1=0 \Rightarrow x=-\frac{1}{2}$$

Therefore, the solutions to the equation are $x=3$ and $x=-\frac{1}{2}$.

Conclusion

In this article, we have explored the steps to solve the equation $\log_3(x+2)=\log_3(2x^2-1)$. We started by setting the arguments of the logarithms equal to each other, then rearranged the equation to form a quadratic equation, and finally factored the quadratic to find the solutions. We found that the solutions to the equation are $x=3$ and $x=-\frac{1}{2}$.

Discussion

The equation $\log_3(x+2)=\log_3(2x^2-1)$ is a classic example of a logarithmic equation. It involves the use of properties of logarithms to simplify and solve the equation. The steps we followed to solve the equation are:

1. Set the arguments of the logarithms equal to each other.
2. Rearrange the equation to form a quadratic equation.
3. Factor the quadratic to find the solutions.

These steps are essential in solving logarithmic equations, and they can be applied to a wide range of problems.

Real-World Applications

The equation $\log_3(x+2)=\log_3(2x^2-1)$ has many real-world applications. For example, it can be used to model population growth, chemical reactions, and electrical circuits. In each of these cases, the equation can be used to find the values of variables that satisfy the equation.

Future Research

There are many areas of research that involve logarithmic equations. For example, researchers are currently studying the properties of logarithmic functions and their applications in machine learning and data analysis. Additionally, researchers are exploring the use of logarithmic equations in modeling complex systems, such as financial markets and climate models.

Conclusion

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Q: What is the first step in solving the equation $\log_3(x+2)=\log_3(2x^2-1)$?

A: The first step in solving the equation is to set the arguments of the logarithms equal to each other. This means that we equate the expressions inside the logarithms:

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

Since the logarithms have the same base, we can drop the logarithms and equate the expressions inside:

$$x+2=2x^2-1$$

Q: Why do we need to set the arguments of the logarithms equal to each other?

A: We need to set the arguments of the logarithms equal to each other because the logarithmic function is one-to-one. This means that if $\log_b(x)=\log_b(y)$, then $x=y$. By setting the arguments of the logarithms equal to each other, we can eliminate the logarithms and solve for the variable.

Q: How do we rearrange the equation to form a quadratic equation?

A: To rearrange the equation to form a quadratic equation, we need to move all the terms to one side of the equation:

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

This is a quadratic equation in the form $ax^2+bx+c=0$, where $a=2$, $b=-1$, and $c=-3$.

Q: How do we factor the quadratic equation?

A: To factor the quadratic equation, we need to find two numbers whose product is $ac$ and whose sum is $b$. In this case, we need to find two numbers whose product is $2(-3)=-6$ and whose sum is $-1$.

After some trial and error, we find that the numbers are $-3$ and $2$, since $(-3)(2)=-6$ and $(-3)+2=-1$. Therefore, we can factor the quadratic as:

$$0=(x-3)(2x+1)$$

Q: How do we find the solutions to the equation?

A: To find the solutions to the equation, we need to set each factor equal to zero and solve for $x$:

$$x-3=0 \Rightarrow x=3$$

$$2x+1=0 \Rightarrow x=-\frac{1}{2}$$

Therefore, the solutions to the equation are $x=3$ and $x=-\frac{1}{2}$.

Q: What are some real-world applications of the equation $\log_3(x+2)=\log_3(2x^2-1)$?

A: The equation $\log_3(x+2)=\log_3(2x^2-1)$ has many real-world applications. For example, it can be used to model population growth, chemical reactions, and electrical circuits. In each of these cases, the equation can be used to find the values of variables that satisfy the equation.

Q: What are some areas of research that involve logarithmic equations?

A: There are many areas of research that involve logarithmic equations. For example, researchers are currently studying the properties of logarithmic functions and their applications in machine learning and data analysis. Additionally, researchers are exploring the use of logarithmic equations in modeling complex systems, such as financial markets and climate models.

Q: How can I apply the steps to solve the equation $\log_3(x+2)=\log_3(2x^2-1)$ to other problems?

A: The steps to solve the equation $\log_3(x+2)=\log_3(2x^2-1)$ can be applied to other problems that involve logarithmic equations. To do this, you need to:

1. Set the arguments of the logarithms equal to each other.
2. Rearrange the equation to form a quadratic equation.
3. Factor the quadratic to find the solutions.

By following these steps, you can solve a wide range of problems that involve logarithmic equations.

Conclusion

In conclusion, the equation $\log_3(x+2)=\log_3(2x^2-1)$ is a classic example of a logarithmic equation. We have explored the steps to solve the equation, including setting the arguments of the logarithms equal to each other, rearranging the equation to form a quadratic equation, and factoring the quadratic to find the solutions. We have also discussed the real-world applications of the equation and the areas of research that involve logarithmic equations.