Solve The Equation For \[$x\$\]:$\[ \log \left(\frac{3x + 1}{2x - 7}\right) = 3 \\]

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Introduction

In this article, we will delve into solving a logarithmic equation involving a fraction. The equation is $\log \left(\frac{3x + 1}{2x - 7}\right) = 3$. We will use various mathematical techniques to isolate the variable $x$ and find its value. This equation is a great example of how logarithms can be used to solve equations involving fractions.

Understanding the Equation

The given equation is $\log \left(\frac{3x + 1}{2x - 7}\right) = 3$. To solve this equation, we need to understand the properties of logarithms. The logarithm of a number is the exponent to which a base number must be raised to produce that number. In this case, the base is not explicitly given, but we can assume it to be 10, as it is the most commonly used base in logarithmic equations.

Step 1: Exponentiate Both Sides

To get rid of the logarithm, we can exponentiate both sides of the equation. This means we can raise the base (10) to the power of both sides of the equation. This gives us:

$$\left(\log \left(\frac{3x + 1}{2x - 7}\right)\right)^{10} = 3^{10}$$

Step 2: Simplify the Equation

Now, we can simplify the equation by canceling out the logarithm. This gives us:

$$\frac{3x + 1}{2x - 7} = 10^3$$

Step 3: Cross Multiply

To get rid of the fraction, we can cross multiply. This means we can multiply both sides of the equation by the denominator $(2x - 7)$. This gives us:

$$(3x + 1)(2x - 7) = 10^3(2x - 7)$$

Step 4: Expand and Simplify

Now, we can expand and simplify the equation. This gives us:

$$6x^2 - 21x + 2x - 7 = 2000x - 7000$$

Step 5: Combine Like Terms

We can combine like terms by adding or subtracting the coefficients of the same variables. This gives us:

$$6x^2 - 19x - 7000 = 2000x - 7000$$

Step 6: Move All Terms to One Side

To isolate the variable $x$, we can move all the terms to one side of the equation. This gives us:

$$6x^2 - 19x - 2000x = 0$$

Step 7: Combine Like Terms

We can combine like terms by adding or subtracting the coefficients of the same variables. This gives us:

$$6x^2 - 2219x = 0$$

Step 8: Factor Out the GCF

To make it easier to solve the equation, we can factor out the greatest common factor (GCF) of the terms. In this case, the GCF is $x$. This gives us:

$$x(6x - 2219) = 0$$

Step 9: Solve for $x$

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

$$x = 0 \quad \text{or} \quad 6x - 2219 = 0$$

Step 10: Solve the Second Equation

To solve the second equation, we can add 2219 to both sides. This gives us:

$$6x = 2219$$

Step 11: Divide Both Sides

Now, we can divide both sides of the equation by 6. This gives us:

$$x = \frac{2219}{6}$$

Conclusion

In this article, we solved the equation $\log \left(\frac{3x + 1}{2x - 7}\right) = 3$ using various mathematical techniques. We started by exponentiating both sides of the equation, then simplified and cross multiplied to get rid of the fraction. We then combined like terms, moved all the terms to one side, and factored out the GCF to make it easier to solve the equation. Finally, we solved for $x$ by setting each factor equal to zero and found two possible solutions: $x = 0$ and $x = \frac{2219}{6}$.

Final Answer

The final answer is $\boxed{\frac{2219}{6}}$.

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Introduction

In our previous article, we solved the equation $\log \left(\frac{3x + 1}{2x - 7}\right) = 3$ using various mathematical techniques. In this article, we will answer some frequently asked questions (FAQs) related to the solution of this equation.

Q: What is the base of the logarithm in the equation?

A: The base of the logarithm in the equation is not explicitly given, but we can assume it to be 10, as it is the most commonly used base in logarithmic equations.

Q: How do I exponentiate both sides of the equation?

A: To exponentiate both sides of the equation, you can raise the base (10) to the power of both sides of the equation. This gives us:

$$\left(\log \left(\frac{3x + 1}{2x - 7}\right)\right)^{10} = 3^{10}$$

Q: Why do I need to cross multiply?

A: You need to cross multiply to get rid of the fraction. This means you can multiply both sides of the equation by the denominator $(2x - 7)$. This gives us:

$$(3x + 1)(2x - 7) = 10^3(2x - 7)$$

Q: How do I combine like terms?

A: To combine like terms, you can add or subtract the coefficients of the same variables. For example, in the equation:

$$6x^2 - 19x - 2000x = 0$$

You can combine the like terms by adding the coefficients of the $x$ terms:

$$6x^2 - 2219x = 0$$

Q: What is the greatest common factor (GCF) of the terms?

A: The greatest common factor (GCF) of the terms is the largest number that divides all the terms without leaving a remainder. In this case, the GCF is $x$. This means you can factor out the GCF from the equation:

$$x(6x - 2219) = 0$$

Q: How do I solve for $x$?

A: To solve for $x$, you can set each factor equal to zero. This gives us:

$$x = 0 \quad \text{or} \quad 6x - 2219 = 0$$

Q: What is the final answer?

A: The final answer is $\boxed{\frac{2219}{6}}$.

Q: Can I use a calculator to solve the equation?

A: Yes, you can use a calculator to solve the equation. However, it's always a good idea to understand the steps involved in solving the equation, as this will help you to verify the solution and understand the underlying mathematics.

Q: What if I get a different solution using a calculator?

A: If you get a different solution using a calculator, it's possible that there is an error in the calculation or that the calculator is not set to the correct mode. In this case, it's always a good idea to double-check your work and verify the solution using a different method.

Conclusion

In this article, we answered some frequently asked questions (FAQs) related to the solution of the equation $\log \left(\frac{3x + 1}{2x - 7}\right) = 3$. We hope that this article has been helpful in clarifying any doubts you may have had about the solution of this equation.