Solve For $x$.$\log _4(2x-3) = 2$$ X = X = X = [/tex]

Introduction

Logarithmic equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will focus on solving a specific type of logarithmic equation, namely the equation involving a logarithm with a base of 4. We will use the given equation $\log _4(2x-3) = 2$ as an example and walk through the steps to solve for $x$.

Understanding Logarithmic Equations

Before we dive into solving the equation, let's take a moment to understand what logarithmic equations are. A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. In other words, if $y = \log _a(x)$, then $a^y = x$. Logarithmic equations can be solved using various methods, including the use of logarithmic properties and the change of base formula.

The Given Equation

The given equation is $\log _4(2x-3) = 2$. This equation involves a logarithm with a base of 4 and can be solved using the properties of logarithms.

Step 1: Apply the Definition of a Logarithm

To solve the equation, we can start by applying the definition of a logarithm. The definition states that if $y = \log _a(x)$, then $a^y = x$. In this case, we have $\log _4(2x-3) = 2$, so we can rewrite the equation as $4^2 = 2x-3$.

Step 2: Simplify the Equation

Now that we have rewritten the equation, we can simplify it by evaluating the exponent. $4^2 = 16$, so the equation becomes $16 = 2x-3$.

Step 3: Add 3 to Both Sides

To isolate the term with the variable, we can add 3 to both sides of the equation. This gives us $16 + 3 = 2x - 3 + 3$, which simplifies to $19 = 2x$.

Step 4: Divide Both Sides by 2

Finally, we can solve for $x$ by dividing both sides of the equation by 2. This gives us $\frac{19}{2} = x$.

Conclusion

In this article, we have walked through the steps to solve a logarithmic equation involving a base of 4. We have applied the definition of a logarithm, simplified the equation, added 3 to both sides, and finally divided both sides by 2 to solve for $x$. The final answer is $\boxed{\frac{19}{2}}$.

Common Mistakes to Avoid

When solving logarithmic equations, there are several common mistakes to avoid. These include:

• Not applying the definition of a logarithm: Failing to apply the definition of a logarithm can lead to incorrect solutions.
• Not simplifying the equation: Failing to simplify the equation can make it difficult to solve.
• Not isolating the term with the variable: Failing to isolate the term with the variable can make it difficult to solve for $x$.

Tips and Tricks

When solving logarithmic equations, there are several tips and tricks to keep in mind. These include:

• Using the change of base formula: The change of base formula can be used to rewrite a logarithmic equation in terms of a common base.
• Applying logarithmic properties: Logarithmic properties, such as the product rule and the quotient rule, can be used to simplify logarithmic equations.
• Using a calculator: A calculator can be used to evaluate logarithmic expressions and simplify equations.

Real-World Applications

Logarithmic equations have numerous real-world applications. These include:

• Finance: Logarithmic equations are used in finance to calculate interest rates and investment returns.
• Science: Logarithmic equations are used in science to model population growth and decay.
• Engineering: Logarithmic equations are used in engineering to design and optimize systems.

Conclusion

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Frequently Asked Questions

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. In other words, if $y = \log _a(x)$, then $a^y = x$.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you can start by applying the definition of a logarithm, simplifying the equation, adding or subtracting the same value to both sides, and finally dividing both sides by the coefficient of the variable.

Q: What is the change of base formula?

A: The change of base formula is a formula that allows you to rewrite a logarithmic equation in terms of a common base. The formula is $\log _a(x) = \frac{\log _b(x)}{\log _b(a)}$.

Q: How do I use the change of base formula?

A: To use the change of base formula, you can substitute the expression $\frac{\log _b(x)}{\log _b(a)}$ for $\log _a(x)$ in the original equation.

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

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

• Not applying the definition of a logarithm
• Not simplifying the equation
• Not isolating the term with the variable
• Not using the change of base formula when necessary

Q: How do I apply logarithmic properties to simplify logarithmic equations?

A: To apply logarithmic properties to simplify logarithmic equations, you can use the product rule, the quotient rule, and the power rule. The product rule states that $\log _a(xy) = \log _a(x) + \log _a(y)$. The quotient rule states that $\log _a(\frac{x}{y}) = \log _a(x) - \log _a(y)$. The power rule states that $\log _a(x^y) = y\log _a(x)$.

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

A: Logarithmic equations have numerous real-world applications, including:

• Finance: Logarithmic equations are used in finance to calculate interest rates and investment returns.
• Science: Logarithmic equations are used in science to model population growth and decay.
• Engineering: Logarithmic equations are used in engineering to design and optimize systems.

Q: How do I use a calculator to evaluate logarithmic expressions and simplify equations?

A: To use a calculator to evaluate logarithmic expressions and simplify equations, you can enter the expression into the calculator and use the logarithmic functions to evaluate the expression.

Q: What are some tips and tricks for solving logarithmic equations?

A: Some tips and tricks for solving logarithmic equations include:

• Using the change of base formula when necessary
• Applying logarithmic properties to simplify the equation
• Using a calculator to evaluate logarithmic expressions and simplify equations
• Checking your work to ensure that the solution is correct

Conclusion

In conclusion, solving logarithmic equations can be a challenging but rewarding task. By applying the definition of a logarithm, simplifying the equation, adding or subtracting the same value to both sides, and finally dividing both sides by the coefficient of the variable, you can solve for $x$. Remember to avoid common mistakes and use tips and tricks to make solving logarithmic equations easier.