Solving Logarithmic Equations Step-by-Step Log_2 X = Log_4 9

Introduction

In the realm of mathematics, logarithmic equations present a fascinating challenge. These equations involve logarithms, which are essentially the inverse operation to exponentiation. Solving them requires a solid understanding of logarithmic properties and the ability to manipulate equations effectively. In this comprehensive guide, we will delve into the process of solving the logarithmic equation $\log _2 x=\log _4 9$. This seemingly simple equation encapsulates several key concepts that are fundamental to mastering logarithmic equations. We will break down each step, providing clear explanations and justifications to ensure a thorough understanding. Whether you are a student grappling with logarithmic equations for the first time or a seasoned mathematician looking to refresh your skills, this guide will provide you with the tools and knowledge necessary to confidently tackle these types of problems. So, let's embark on this mathematical journey and unravel the solution to $\log _2 x=\log _4 9$.

Understanding Logarithms

Before we dive into solving the equation, let's first establish a clear understanding of logarithms. A logarithm answers the question: "To what power must we raise a base to get a certain number?" In mathematical terms, if $b^y = x$, then the logarithm of $x$ to the base $b$ is $y$, written as $\log_b x = y$. Here, $b$ is the base, $x$ is the argument, and $y$ is the exponent. It's crucial to grasp this fundamental relationship between logarithms and exponents, as it forms the bedrock for solving logarithmic equations.

For instance, $\log_2 8 = 3$ because $2^3 = 8$. Similarly, $\log_{10} 100 = 2$ because $10^2 = 100$. The base of the logarithm plays a vital role in determining the value of the logarithm. Different bases lead to different logarithmic values for the same argument. This is why it's essential to pay close attention to the base when working with logarithmic equations. In our equation, we have logarithms with bases 2 and 4, which highlights the need for a method to reconcile these different bases.

The properties of logarithms are essential tools for simplifying and solving logarithmic equations. Some key properties include:

• Product Rule: $\log_b (mn) = \log_b m + \log_b n$
• Quotient Rule: $\log_b (m/n) = \log_b m - \log_b n$
• Power Rule: $\log_b (m^p) = p \log_b m$
• Change of Base Formula: $\log_b a = \frac{\log_c a}{\log_c b}$

These properties allow us to manipulate logarithmic expressions, combine or separate logarithms, and change the base of a logarithm, all of which are crucial techniques for solving logarithmic equations. In the context of our equation, the change of base formula will be particularly useful in expressing both logarithms in the same base, allowing us to compare and equate them effectively.

Applying the Change of Base Formula

The change of base formula is a cornerstone technique when dealing with logarithmic equations involving different bases. This formula states that $\log_b a = \frac{\log_c a}{\log_c b}$, where $a$, $b$, and $c$ are positive numbers and $b$ and $c$ are not equal to 1. The power of this formula lies in its ability to transform a logarithm from one base to another, providing a common ground for comparison and manipulation. In our equation, $\log _2 x=\log _4 9$, we have logarithms with bases 2 and 4. To solve this equation, we need to express both logarithms in the same base. We can choose either base 2 or base 4, or even a common base like 10 or $e$ (natural logarithm). However, choosing base 2 seems more intuitive in this case, as 4 is a power of 2.

Let's apply the change of base formula to the right side of the equation, $\log_4 9$. We want to convert this logarithm to base 2. Using the change of base formula, we have:

$\log_4 9 = \frac{\log_2 9}{\log_2 4}$

Now, we know that $\log_2 4 = 2$ because $2^2 = 4$. Substituting this into the equation, we get:

$\log_4 9 = \frac{\log_2 9}{2}$

This transformation is a crucial step in solving the equation, as it allows us to express both sides of the equation in terms of logarithms with the same base. Now our original equation $\log _2 x=\log _4 9$ becomes:

$\log_2 x = \frac{\log_2 9}{2}$

This new form of the equation is much easier to work with. We have successfully used the change of base formula to unify the bases of the logarithms, paving the way for further simplification and ultimately solving for $x$.

Simplifying the Equation

With the equation now in the form $\log_2 x = \frac{\log_2 9}{2}$, we can proceed to simplify it further. Our goal is to isolate $x$ and determine its value. To do this, we need to manipulate the equation in a way that eliminates the logarithm on the left side. One approach is to utilize the power rule of logarithms in reverse. Recall that the power rule states $\log_b (m^p) = p \log_b m$. We can use this rule to rewrite the right side of our equation.

$\frac{\log_2 9}{2}$ can be rewritten as $\frac{1}{2} \log_2 9$. Now, applying the power rule in reverse, we can bring the $\frac{1}{2}$ inside the logarithm as an exponent:

$\frac{1}{2} \log_2 9 = \log_2 (9^{\frac{1}{2}})$

Remember that raising a number to the power of $\frac{1}{2}$ is equivalent to taking the square root of that number. Therefore, $9^{\frac{1}{2}} = \sqrt{9} = 3$. Substituting this back into our equation, we have:

$\log_2 x = \log_2 3$

This simplified form of the equation is a significant step forward. We now have a logarithm on both sides of the equation with the same base. This allows us to equate the arguments of the logarithms, leading us directly to the solution for $x$.

Solving for x

Having simplified the equation to $\log_2 x = \log_2 3$, we are now in the final stage of solving for x. The beauty of this form is that it directly reveals the value of $x$. If the logarithms of two numbers to the same base are equal, then the numbers themselves must be equal. This is a fundamental property of logarithms that stems from the fact that the logarithmic function is one-to-one.

Therefore, from the equation $\log_2 x = \log_2 3$, we can directly conclude that:

$x = 3$

This is the solution to the logarithmic equation $\log _2 x=\log _4 9$. It's a concise and elegant solution that demonstrates the power of logarithmic properties and the change of base formula.

To ensure the validity of our solution, it's always a good practice to substitute the value of $x$ back into the original equation and verify that it holds true. In this case, we substitute $x = 3$ into the original equation:

$\log _2 3=\log _4 9$

We already know that $\log_4 9 = \frac{\log_2 9}{2}$. And we simplified $\frac{\log_2 9}{2}$ to $\log_2 (9^{\frac{1}{2}}) = \log_2 3$. So, the equation holds true.

This verification step provides confidence in our solution and reinforces the understanding of the logarithmic principles involved. We have successfully solved the logarithmic equation and confirmed the validity of our answer.

Conclusion

In this comprehensive guide, we have meticulously walked through the process of solving the logarithmic equation $\log _2 x=\log _4 9$. We began by establishing a solid understanding of logarithms, their properties, and the crucial relationship between logarithms and exponents. We then introduced the change of base formula, a powerful tool for manipulating logarithms with different bases. By applying this formula, we transformed the original equation into a more manageable form with a common base.

We further simplified the equation using the power rule of logarithms, rewriting the equation in a way that directly revealed the solution for $x$. Finally, by equating the arguments of the logarithms, we arrived at the solution $x = 3$. To ensure the accuracy of our solution, we performed a verification step, substituting the value of $x$ back into the original equation and confirming that it holds true.

This step-by-step approach highlights the importance of understanding the underlying principles of logarithms and the strategic application of logarithmic properties. Solving logarithmic equations is not just about memorizing formulas; it's about developing a deep conceptual understanding and the ability to manipulate equations effectively. The techniques and insights gained from solving this equation can be applied to a wide range of logarithmic problems.

Mastering logarithmic equations opens doors to more advanced mathematical concepts and applications. Logarithms are used extensively in various fields, including science, engineering, and finance. A strong foundation in logarithms is essential for anyone pursuing these disciplines. By diligently practicing and applying the principles outlined in this guide, you can confidently tackle logarithmic equations and unlock the power of this fundamental mathematical concept.