Solving Logarithmic Equations Step By Step Guide

Introduction

In the realm of mathematics, logarithmic equations often present a fascinating challenge. Understanding the properties of logarithms is crucial in solving these equations effectively. This article delves into the solution of the logarithmic equation ${\log _3(4 x)-2 \log _3 x=2}$, providing a detailed, step-by-step approach to arrive at the correct answer. Our exploration will not only focus on the mechanics of solving the equation but also on the underlying principles that govern logarithmic operations. Mastering these principles is essential for anyone looking to excel in algebra and calculus, as logarithmic equations frequently appear in various mathematical contexts.

Understanding Logarithms

Before we dive into the solution, let's briefly recap the basics of logarithms. A logarithm is the inverse operation to exponentiation. The logarithmic expression ${\log_b a = c}$ can be rewritten in its exponential form as ${b^c = a}$. Here, ${b}$ is the base of the logarithm, ${a}$ is the argument, and ${c}$ is the exponent. This fundamental relationship is key to manipulating and solving logarithmic equations. Logarithms help simplify complex calculations, especially when dealing with very large or very small numbers. They are widely used in fields such as physics, engineering, computer science, and economics for modeling and solving problems involving exponential growth and decay.

Breaking Down the Equation

The given equation is ${\log _3(4 x)-2 \log _3 x=2}$. To solve this equation, we will leverage the properties of logarithms to simplify the expression and isolate the variable ${x}$. Our approach will involve using the power rule of logarithms, the quotient rule, and converting the logarithmic equation into its exponential form. Each step is carefully explained to ensure clarity and understanding. By following this structured method, you will not only solve this particular equation but also gain the skills to tackle similar logarithmic problems.

Step 1: Applying the Power Rule of Logarithms

The power rule of logarithms states that ${\log_b(a^c) = c \log_b(a)}$. We can use this rule to simplify the term ${2 \log _3 x}$ in our equation. Applying the power rule, we rewrite ${2 \log _3 x}$ as ${\log _3(x^2)}$. This transformation simplifies the equation, making it easier to manipulate further. The power rule is a fundamental tool in simplifying logarithmic expressions and is crucial for solving equations involving logarithms.

Step 2: Rewriting the Equation

Now, our equation looks like this: ${\log _3(4 x) - \log _3(x^2) = 2}$. We have successfully applied the power rule to eliminate the coefficient in front of the logarithmic term. This step brings us closer to combining the logarithmic terms and simplifying the equation. The ability to rewrite and simplify logarithmic expressions is a key skill in solving logarithmic equations.

Step 3: Applying the Quotient Rule of Logarithms

The quotient rule of logarithms states that ${\log_b(a) - \log_b(c) = \log_b(\frac{a}{c})}$. We can apply this rule to combine the two logarithmic terms on the left side of our equation. Using the quotient rule, we rewrite ${\log _3(4 x) - \log _3(x^2)}$ as ${\log _3(\frac{4 x}{x^2})}$. This step significantly simplifies the equation by reducing the number of logarithmic terms. The quotient rule is another essential tool in simplifying logarithmic expressions.

Step 4: Simplifying the Fraction

After applying the quotient rule, our equation is now ${\log _3(\frac{4 x}{x^2}) = 2}$. We can simplify the fraction inside the logarithm. The expression ${\frac{4 x}{x^2}}$ simplifies to ${\frac{4}{x}}$, assuming ${x \neq 0}$. This simplification makes the equation easier to convert into its exponential form. Simplifying algebraic expressions is a crucial step in solving many mathematical problems.

Converting to Exponential Form

Step 5: Converting the Logarithmic Equation to Exponential Form

Our simplified equation is ${\log _3(\frac{4}{x}) = 2}$. To eliminate the logarithm, we convert the equation to its exponential form. Recall that ${\log_b a = c}$ is equivalent to ${b^c = a}$. Applying this to our equation, we get ${3^2 = \frac{4}{x}}$. This conversion transforms the logarithmic equation into a more manageable algebraic equation. Understanding the relationship between logarithmic and exponential forms is essential for solving logarithmic equations.

Solving for x

Step 6: Solving for x

Now we have the equation ${3^2 = \frac{4}{x}}$. We simplify ${3^2}$ to ${9}$, so the equation becomes ${9 = \frac{4}{x}}$. To solve for ${x}$, we multiply both sides of the equation by ${x}$ to get ${9x = 4}$. Then, we divide both sides by ${9}$ to isolate ${x}$. This gives us ${x = \frac{4}{9}}$. Solving for variables in algebraic equations is a fundamental skill in mathematics.

Verifying the Solution

Step 7: Verifying the Solution

It is crucial to verify that our solution, ${x = \frac{4}{9}}$, is valid by substituting it back into the original equation. The original equation is ${\log _3(4 x)-2 \log _3 x=2}$. Substituting ${x = \frac{4}{9}}$ into the equation, we get:

${\log _3(4 \cdot \frac{4}{9})-2 \log _3(\frac{4}{9}) = 2}$

Let's simplify this expression step by step:

${\log _3(\frac{16}{9})-2 \log _3(\frac{4}{9}) = 2}$

We can rewrite ${\log _3(\frac{16}{9})}$ as ${\log _3(\frac{4^2}{3^2})}$ and ${\log _3(\frac{4}{9})}$ as ${\log _3(\frac{4}{3^2})}$. Applying the power rule, we get:

${\log _3((\frac{4}{3})^2) - 2 \log _3(\frac{4}{9}) = 2}$

${2 \log _3(\frac{4}{3}) - 2 \log _3(\frac{4}{9}) = 2}$

Now, we can factor out the 2:

${2 [\log _3(\frac{4}{3}) - \log _3(\frac{4}{9})] = 2}$

Divide both sides by 2:

${\log _3(\frac{4}{3}) - \log _3(\frac{4}{9}) = 1}$

Applying the quotient rule, we get:

${\log _3(\frac{\frac{4}{3}}{\frac{4}{9}}) = 1}$

${\log _3(\frac{4}{3} \cdot \frac{9}{4}) = 1}$

${\log _3(3) = 1}$

Since ${\log _3(3) = 1}$, our equation simplifies to:

${1 = 1}$

This confirms that our solution, ${x = \frac{4}{9}}$, is indeed correct. Verifying solutions is a critical step in problem-solving to ensure accuracy.

Conclusion

In conclusion, the solution to the equation ${\log _3(4 x)-2 \log _3 x=2}$ is ${x = \frac{4}{9}}$. This solution was obtained by systematically applying the power and quotient rules of logarithms, converting the logarithmic equation to exponential form, and solving the resulting algebraic equation. Remember, mastering the properties of logarithms is key to solving logarithmic equations effectively. By understanding and applying these principles, you can confidently tackle a wide range of logarithmic problems in mathematics and related fields.

Final Answer: A. $\frac{4}{9}$