Solving Exponential Equations Find X In 2(3^x) = 5

In the realm of mathematical equations, exponential equations hold a significant place, often posing intriguing challenges for students and enthusiasts alike. Exponential equations, characterized by a variable appearing in the exponent, require specific techniques to unravel their solutions. This article delves into the step-by-step process of solving the equation $2(3^x) = 5$, elucidating the underlying principles and providing a comprehensive understanding of the methods involved. Let's embark on this mathematical journey to find the elusive value of x.

Before diving into the solution, it's crucial to grasp the essence of exponential equations. Unlike linear or polynomial equations where the variable typically resides in the base, exponential equations feature the variable as an exponent. This positioning fundamentally alters the approach to solving them. The equation $2(3^x) = 5$ exemplifies this, where x is the exponent of the base 3. To effectively tackle such equations, we often employ logarithms, the inverse operation of exponentiation. Logarithms allow us to "bring down" the exponent, transforming the equation into a more manageable form. This process involves understanding logarithmic properties and applying them judiciously to isolate the variable.

Step 1 Isolating the Exponential Term

The initial step in solving $2(3^x) = 5$ involves isolating the exponential term, which in this case is $3^x$. To achieve this, we divide both sides of the equation by 2:

$2(3^x) / 2 = 5 / 2$

This simplifies to:

$3^x = 2.5$

Now, the exponential term is isolated, paving the way for the application of logarithms.

Step 2 Applying Logarithms

The next crucial step is to apply logarithms to both sides of the equation. The choice of logarithm base is often a matter of convenience, with the common logarithm (base 10) and the natural logarithm (base e) being popular choices. For this example, we'll use the natural logarithm (ln) due to its widespread use and calculator availability. Applying the natural logarithm to both sides gives us:

ln($3^x$) = ln(2.5)

Step 3 Utilizing Logarithmic Properties

One of the fundamental properties of logarithms states that ln($a^b$) = b * ln(a). Applying this property to the left side of the equation allows us to bring the exponent x down as a coefficient:

x * ln(3) = ln(2.5)

This transformation is pivotal, as it converts the exponential equation into a linear equation, which is far easier to solve.

Step 4 Isolating x

To isolate x, we divide both sides of the equation by ln(3):

x = ln(2.5) / ln(3)

This step isolates x, expressing it in terms of logarithms of known quantities.

Step 5 Calculating the Value of x

The final step involves using a calculator to evaluate the expression ln(2.5) / ln(3). The natural logarithm of 2.5 is approximately 0.9163, and the natural logarithm of 3 is approximately 1.0986. Therefore:

x ≈ 0.9163 / 1.0986

x ≈ 0.8341

Thus, the solution to the equation $2(3^x) = 5$ is approximately 0.8341.

While we used the natural logarithm in our solution, the common logarithm (log base 10) can also be employed. The process is analogous, with the only difference being the base of the logarithm. Applying the common logarithm to both sides of $3^x = 2.5$ yields:

log($3^x$) = log(2.5)

Using the logarithmic property, we get:

x * log(3) = log(2.5)

Isolating x, we have:

x = log(2.5) / log(3)

Using a calculator, log(2.5) ≈ 0.3979 and log(3) ≈ 0.4771. Therefore:

x ≈ 0.3979 / 0.4771

x ≈ 0.8341

As expected, the solution obtained using the common logarithm is the same as that obtained using the natural logarithm, demonstrating the versatility of logarithmic methods.

To ensure the accuracy of our solution, it's prudent to verify it by substituting the calculated value of x back into the original equation:

$2(3^x) = 5$

Substituting x ≈ 0.8341, we get:

$2(3^{0.8341}) ≈ 5$

Using a calculator, $3^{0.8341}$ ≈ 2.5, so:

2(2.5) ≈ 5

5 ≈ 5

This confirms that our solution is indeed correct.

When solving exponential equations, several common pitfalls can lead to incorrect solutions. One frequent mistake is attempting to directly solve for x without isolating the exponential term first. For instance, in the equation $2(3^x) = 5$, some might erroneously try to take the logarithm of the entire left side without dividing by 2 first. Another common error is misapplying logarithmic properties, such as incorrectly distributing the logarithm over a sum or difference. It's crucial to remember that ln(a + b) ≠ ln(a) + ln(b) and ln(a - b) ≠ ln(a) - ln(b). Careful attention to these details is essential for accurate problem-solving.

Exponential equations are not merely abstract mathematical constructs; they have profound applications in various fields, including physics, engineering, finance, and computer science. In physics, they model radioactive decay and population growth. In finance, they are used to calculate compound interest and investment returns. In computer science, they appear in algorithms and data structures, such as binary trees and search algorithms. Understanding how to solve exponential equations is therefore not only a valuable mathematical skill but also a crucial tool for tackling real-world problems.

In conclusion, solving the exponential equation $2(3^x) = 5$ involves a systematic application of logarithmic principles. By isolating the exponential term, applying logarithms, utilizing logarithmic properties, and isolating the variable, we successfully determined that x ≈ 0.8341. This process underscores the power of logarithms in unraveling exponential relationships. Moreover, the ability to solve exponential equations extends far beyond the classroom, providing a foundation for understanding and modeling phenomena in diverse scientific and technological domains. Mastery of these techniques empowers individuals to tackle complex problems and make informed decisions in a world increasingly shaped by exponential growth and decay.