Solve The Equation: Log ⁡ 4 ( 3 X − 1 ) − Log ⁡ 4 2 = 2 \log_4(3x-1) - \log_4 2 = 2 Lo G 4 ​ ( 3 X − 1 ) − Lo G 4 ​ 2 = 2 Use The Numeric Grid To Type The Solution.

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Introduction

In this article, we will be solving a logarithmic equation involving base 4 logarithms. The equation is $\log_4(3x-1) - \log_4 2 = 2$. We will use the properties of logarithms to simplify the equation and solve for the variable x.

Understanding Logarithms

Before we dive into solving the equation, let's briefly review the concept of logarithms. A logarithm is the inverse operation of exponentiation. In other words, if $a^b = c$, then $\log_a c = b$. The logarithm of a number is the exponent to which a base number must be raised to produce that number.

Properties of Logarithms

There are several properties of logarithms that we will use to simplify the equation. These properties include:

• Product Rule: $\log_a (xy) = \log_a x + \log_a y$
• Quotient Rule: $\log_a \frac{x}{y} = \log_a x - \log_a y$
• Power Rule: $\log_a x^b = b \log_a x$

Simplifying the Equation

Using the properties of logarithms, we can simplify the equation as follows:

$\log_4(3x-1) - \log_4 2 = 2$

Using the Quotient Rule, we can rewrite the equation as:

$\log_4 \frac{3x-1}{2} = 2$

Exponentiating Both Sides

To get rid of the logarithm, we can exponentiate both sides of the equation. Since the base of the logarithm is 4, we can raise 4 to both sides of the equation:

$4^{\log_4 \frac{3x-1}{2}} = 4^2$

Simplifying the Exponents

Using the property of exponents that states $a^{\log_a x} = x$, we can simplify the left-hand side of the equation:

$\frac{3x-1}{2} = 16$

Solving for x

To solve for x, we can multiply both sides of the equation by 2:

$3x-1 = 32$

Adding 1 to both sides of the equation gives us:

$3x = 33$

Dividing both sides of the equation by 3 gives us:

$x = \frac{33}{3}$

$x = 11$

Conclusion

In this article, we solved the equation $\log_4(3x-1) - \log_4 2 = 2$ using the properties of logarithms. We simplified the equation using the Quotient Rule and exponentiated both sides to get rid of the logarithm. Finally, we solved for x by multiplying both sides of the equation by 2 and dividing both sides by 3. The solution to the equation is x = 11.

Step-by-Step Solution

Here is the step-by-step solution to the equation:

1. $\log_4(3x-1) - \log_4 2 = 2$
2. $\log_4 \frac{3x-1}{2} = 2$
3. $4^{\log_4 \frac{3x-1}{2}} = 4^2$
4. $\frac{3x-1}{2} = 16$
5. $3x-1 = 32$
6. $3x = 33$
7. $x = \frac{33}{3}$
8. $x = 11$

Final Answer

The final answer is $\boxed{11}$.

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Introduction

In our previous article, we solved the equation $\log_4(3x-1) - \log_4 2 = 2$ using the properties of logarithms. In this article, we will answer some frequently asked questions about the solution to the equation.

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

A: The base of the logarithm in the equation is 4.

Q: What is the property of logarithms used to simplify the equation?

A: The Quotient Rule is used to simplify the equation. The Quotient Rule states that $\log_a \frac{x}{y} = \log_a x - \log_a y$.

Q: How do you get rid of the logarithm in the equation?

A: To get rid of the logarithm, we can exponentiate both sides of the equation. Since the base of the logarithm is 4, we can raise 4 to both sides of the equation.

Q: What is the final answer to the equation?

A: The final answer to the equation is x = 11.

Q: Can you explain the step-by-step solution to the equation?

A: Here is the step-by-step solution to the equation:

1. $\log_4(3x-1) - \log_4 2 = 2$
2. $\log_4 \frac{3x-1}{2} = 2$
3. $4^{\log_4 \frac{3x-1}{2}} = 4^2$
4. $\frac{3x-1}{2} = 16$
5. $3x-1 = 32$
6. $3x = 33$
7. $x = \frac{33}{3}$
8. $x = 11$

Q: What is the significance of the solution to the equation?

A: The solution to the equation is x = 11, which means that when the base 4 logarithm of (3x-1) is subtracted by the base 4 logarithm of 2, the result is 2.

Q: Can you provide more examples of logarithmic equations?

A: Yes, here are a few more examples of logarithmic equations:

• $\log_2(x+1) - \log_2 3 = 1$
• $\log_5(2x-1) + \log_5 2 = 3$
• $\log_3(x-1) - \log_3 2 = 2$

Q: How do you solve logarithmic equations with different bases?

A: To solve logarithmic equations with different bases, you can use the change of base formula, which states that $\log_a x = \frac{\log_b x}{\log_b a}$.

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

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

• Not using the correct property of logarithms
• Not exponentiating both sides of the equation
• Not simplifying the equation correctly
• Not checking the domain of the logarithm function

Conclusion

In this article, we answered some frequently asked questions about the solution to the equation $\log_4(3x-1) - \log_4 2 = 2$. We also provided some additional examples of logarithmic equations and discussed some common mistakes to avoid when solving logarithmic equations.

Final Answer

The final answer is $\boxed{11}$.