Kim Solved The Equation Below By Graphing A System Of Equations.$\log _2(3 X-1)=\log _4(x+8$\]What Is The Approximate Solution To The Equation?A. 0.6 B. 0.9 C. 1.4 D. 1.6

Introduction

Logarithmic equations can be challenging to solve, especially when they involve different bases. However, with the help of graphing, we can visualize the solution and find the approximate value of the variable. In this article, we will explore how to solve the equation $\log _2(3 x-1)=\log _4(x+8)$ using graphing.

Understanding Logarithmic Equations

Before we dive into the solution, let's review the properties of logarithmic equations. A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. The general form of a logarithmic equation is $\log _a(x) = y$, where $a$ is the base of the logarithm and $x$ is the argument.

The Given Equation

The given equation is $\log _2(3 x-1)=\log _4(x+8)$. To solve this equation, we need to find the value of $x$ that satisfies both sides of the equation.

Step 1: Convert the Equation to Exponential Form

To solve the equation, we can start by converting it to exponential form. We can do this by using the property of logarithms that states $\log _a(x) = y \iff a^y = x$. Applying this property to both sides of the equation, we get:

$$2^{\log _2(3 x-1)} = 4^{\log _4(x+8)}$$

Step 2: Simplify the Equation

Now that we have the equation in exponential form, we can simplify it by using the property of exponents that states $a^{\log _a(x)} = x$. Applying this property to both sides of the equation, we get:

$$3 x-1 = (x+8)^{\frac{1}{2}}$$

Step 3: Graph the Equations

To find the solution to the equation, we can graph the two equations on the same coordinate plane. The first equation is $y = 3 x-1$, and the second equation is $y = (x+8)^{\frac{1}{2}}$. By graphing these equations, we can find the point of intersection, which represents the solution to the equation.

Graphing the First Equation

The first equation is $y = 3 x-1$. To graph this equation, we can start by finding the x-intercept, which is the point where the graph intersects the x-axis. To find the x-intercept, we can set $y = 0$ and solve for $x$. This gives us:

$$0 = 3 x-1 \Rightarrow x = \frac{1}{3}$$

Graphing the Second Equation

The second equation is $y = (x+8)^{\frac{1}{2}}$. To graph this equation, we can start by finding the x-intercept, which is the point where the graph intersects the x-axis. To find the x-intercept, we can set $y = 0$ and solve for $x$. This gives us:

$$0 = (x+8)^{\frac{1}{2}} \Rightarrow x = -8$$

Finding the Point of Intersection

Now that we have graphed both equations, we can find the point of intersection, which represents the solution to the equation. To find the point of intersection, we can use the graphing calculator to find the approximate value of the x-coordinate.

Approximate Solution

Using the graphing calculator, we find that the approximate value of the x-coordinate is $x \approx 1.4$. Therefore, the approximate solution to the equation is $x \approx 1.4$.

Conclusion

In this article, we have shown how to solve the equation $\log _2(3 x-1)=\log _4(x+8)$ using graphing. We started by converting the equation to exponential form, then simplified it by using the property of exponents. Finally, we graphed the two equations on the same coordinate plane and found the point of intersection, which represents the solution to the equation. The approximate solution to the equation is $x \approx 1.4$.

Answer

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Q: What is the main concept behind solving logarithmic equations through graphing?

A: The main concept behind solving logarithmic equations through graphing is to visualize the solution and find the approximate value of the variable. By graphing the two equations on the same coordinate plane, we can find the point of intersection, which represents the solution to the equation.

Q: How do I convert a logarithmic equation to exponential form?

A: To convert a logarithmic equation to exponential form, we can use the property of logarithms that states $\log _a(x) = y \iff a^y = x$. This means that we can rewrite the logarithmic equation as an exponential equation by raising the base to the power of the logarithm.

Q: What is the property of exponents that I can use to simplify the equation?

A: The property of exponents that we can use to simplify the equation is $a^{\log _a(x)} = x$. This means that we can rewrite the exponential equation as a simple equation by raising the base to the power of the logarithm.

Q: How do I graph the two equations on the same coordinate plane?

A: To graph the two equations on the same coordinate plane, we can use a graphing calculator or a graphing software. We can start by graphing the first equation, then graph the second equation on the same coordinate plane. By graphing both equations, we can find the point of intersection, which represents the solution to the equation.

Q: What is the approximate solution to the equation $\log _2(3 x-1)=\log _4(x+8)$?

A: The approximate solution to the equation $\log _2(3 x-1)=\log _4(x+8)$ is $x \approx 1.4$.

Q: Can I use graphing to solve any type of logarithmic equation?

A: Yes, you can use graphing to solve any type of logarithmic equation. However, the equation must be in a form that can be graphed, such as a linear or quadratic equation.

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

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

• Not converting the logarithmic equation to exponential form
• Not simplifying the equation using the property of exponents
• Not graphing the two equations on the same coordinate plane
• Not finding the point of intersection, which represents the solution to the equation

Q: How can I practice solving logarithmic equations through graphing?

A: You can practice solving logarithmic equations through graphing by working through examples and exercises. You can also use online resources, such as graphing calculators and software, to help you visualize the solution.

Q: What are some real-world applications of solving logarithmic equations through graphing?

A: Some real-world applications of solving logarithmic equations through graphing include:

• Modeling population growth and decay
• Analyzing financial data and predicting stock prices
• Solving problems in physics and engineering, such as calculating the trajectory of a projectile or the stress on a material

Conclusion

Solving logarithmic equations through graphing is a powerful tool that can be used to find the approximate solution to a wide range of equations. By understanding the concepts and techniques behind graphing, you can apply this method to solve real-world problems and gain a deeper understanding of logarithmic equations.