Solving Logarithmic Equations Finding The Value Of X

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In this comprehensive exploration, we embark on a journey to solve the intricate logarithmic equation $\log _{\frac{3}{4}} 25=3 x-1$. Our primary objective is to determine the approximate value of $x$ that satisfies this equation. To achieve this, we will meticulously dissect the equation, employing a combination of logarithmic properties and algebraic manipulations. This step-by-step approach will not only lead us to the solution but also provide a clear understanding of the underlying mathematical principles.

Understanding Logarithmic Equations

Logarithmic equations form a cornerstone of mathematical analysis, particularly in fields like calculus, physics, and engineering. At its core, a logarithm answers the fundamental question: To what power must we raise a specific base to obtain a desired number? In the given equation, $\log {\frac{3}{4}} 25=3 x-1$, we encounter a logarithm with a fractional base, which introduces an intriguing layer of complexity. Before diving into the solution, it's crucial to grasp the essence of logarithms and their properties. The logarithm of a number to a given base is the exponent to which the base must be raised to produce that number. For instance, $\log{2} 8 = 3$ because $2^3 = 8$. Understanding this fundamental concept is the key to unraveling the intricacies of logarithmic equations.

Key Properties of Logarithms

To effectively tackle logarithmic equations, a firm grasp of logarithmic properties is indispensable. These properties serve as the bedrock for simplifying and manipulating logarithmic expressions, paving the way for solutions. Here are some essential properties that we will utilize:

  1. Change of Base Formula: This formula enables us to convert logarithms from one base to another. It is particularly useful when dealing with logarithms with unconventional bases, such as the fractional base in our equation. The formula is expressed as:

    log⁡ab=log⁡cblog⁡ca\log_{a} b = \frac{\log_{c} b}{\log_{c} a}

    where $a$, $b$, and $c$ are positive numbers and $a$ and $c$ are not equal to 1.
  2. Power Rule: This rule allows us to simplify logarithms of numbers raised to a power. It states that:

    log⁡abc=clog⁡ab\log_{a} b^c = c \log_{a} b

    where $a$ is a positive number not equal to 1, and $b$ is a positive number.
  3. Logarithm of a Quotient: This property helps us handle logarithms of fractions. It states that:

    log⁡abc=log⁡ab−log⁡ac\log_{a} \frac{b}{c} = \log_{a} b - \log_{a} c

    where $a$, $b$, and $c$ are positive numbers and $a$ is not equal to 1.

By skillfully applying these properties, we can transform complex logarithmic equations into simpler, more manageable forms.

Solving the Equation Step-by-Step

Now, let's embark on the journey of solving the given equation, $\log _{\frac{3}{4}} 25=3 x-1$. We will navigate through the solution process step by step, illuminating each stage with clear explanations and justifications.

Step 1: Applying the Change of Base Formula

The initial challenge lies in dealing with the logarithm with a fractional base, $\frac{3}{4}$. To overcome this, we employ the change of base formula. This powerful tool allows us to convert the logarithm to a more convenient base, such as the natural logarithm (base $e$) or the common logarithm (base 10). For simplicity, we will opt for the common logarithm (base 10). Applying the change of base formula, we get:

log⁡3425=log⁡1025log⁡1034\log _{\frac{3}{4}} 25 = \frac{\log_{10} 25}{\log_{10} \frac{3}{4}}

This transformation lays the foundation for further simplification.

Step 2: Simplifying the Logarithm of a Quotient

Next, we encounter the logarithm of a fraction in the denominator, $\log_{10} \frac{3}{4}$. To simplify this, we invoke the logarithm of a quotient property. This property elegantly expresses the logarithm of a fraction as the difference of the logarithms of the numerator and denominator. Applying this property, we obtain:

log⁡1034=log⁡103−log⁡104\log_{10} \frac{3}{4} = \log_{10} 3 - \log_{10} 4

This simplification further refines the equation, bringing us closer to the solution.

Step 3: Substituting and Rearranging the Equation

Now, we substitute the simplified expression back into the original equation. This yields:

log⁡1025log⁡103−log⁡104=3x−1\frac{\log_{10} 25}{\log_{10} 3 - \log_{10} 4} = 3x - 1

To isolate $x$, we begin by adding 1 to both sides of the equation:

log⁡1025log⁡103−log⁡104+1=3x\frac{\log_{10} 25}{\log_{10} 3 - \log_{10} 4} + 1 = 3x

Step 4: Isolating $x$

The final step in isolating $x$ involves dividing both sides of the equation by 3:

x=13(log⁡1025log⁡103−log⁡104+1)x = \frac{1}{3} \left( \frac{\log_{10} 25}{\log_{10} 3 - \log_{10} 4} + 1 \right)

This expression provides us with the exact value of $x$. However, to obtain a numerical approximation, we need to evaluate the logarithms.

Step 5: Evaluating Logarithms and Approximating $x$

To obtain a numerical approximation of $x$, we utilize a calculator to evaluate the common logarithms. We find that:

  • log⁡1025≈1.3979\log_{10} 25 \approx 1.3979

  • log⁡103≈0.4771\log_{10} 3 \approx 0.4771

  • log⁡104≈0.6021\log_{10} 4 \approx 0.6021

Substituting these values into the expression for $x$, we get:

x≈13(1.39790.4771−0.6021+1)x \approx \frac{1}{3} \left( \frac{1.3979}{0.4771 - 0.6021} + 1 \right)

x≈13(1.3979−0.125+1)x \approx \frac{1}{3} \left( \frac{1.3979}{-0.125} + 1 \right)

x≈13(−11.1832+1)x \approx \frac{1}{3} (-11.1832 + 1)

x≈13(−10.1832)x \approx \frac{1}{3} (-10.1832)

x≈−3.3944x \approx -3.3944

Therefore, the approximate value of $x$ in the equation $\log _{\frac{3}{4}} 25=3 x-1$ is approximately -3.3944. This meticulous step-by-step solution showcases the power of logarithmic properties and algebraic manipulations in unraveling mathematical equations.

Conclusion

In this comprehensive exploration, we have successfully determined the approximate value of $x$ in the logarithmic equation $\log _{\frac{3}{4}} 25=3 x-1$. By employing a combination of logarithmic properties, including the change of base formula and the logarithm of a quotient property, we systematically simplified the equation and isolated $x$. The approximate value of $x$ was found to be -3.3944. This exercise underscores the importance of understanding and applying logarithmic principles in solving mathematical problems. The ability to manipulate logarithmic expressions and equations is a valuable asset in various scientific and engineering disciplines.

The journey through this equation has not only provided us with a numerical solution but also illuminated the elegance and power of mathematical tools. Logarithms, often perceived as abstract concepts, find practical applications in diverse fields, ranging from measuring the intensity of earthquakes to modeling population growth. By mastering the art of solving logarithmic equations, we equip ourselves with a versatile skillset that can be applied to a wide spectrum of real-world scenarios. The satisfaction of unraveling a complex equation and arriving at a precise solution is a testament to the beauty and power of mathematics.

In conclusion, the approximate value of $x$ in the equation $\log _{\frac{3}{4}} 25=3 x-1$ is approximately -3.3944. This solution was obtained through a methodical application of logarithmic properties and algebraic manipulations, highlighting the fundamental role of these tools in mathematical problem-solving. The understanding of logarithms and their applications extends far beyond the realm of academia, playing a crucial role in various scientific and technological endeavors. As we continue to explore the vast landscape of mathematics, the ability to solve equations like this will undoubtedly prove invaluable in our pursuit of knowledge and understanding.