Solving The Exponential Equation -7 * 3^(0.25x) = -10

Introduction

In this article, we will walk through the steps to solve the exponential equation $-7 imes 3^{0.25x} = -10$. Exponential equations are a fundamental topic in algebra and calculus, and mastering their solution techniques is crucial for various applications in mathematics, physics, engineering, and finance. We will explore how to isolate the exponential term, apply logarithms, and use logarithmic properties to find the solution. This detailed explanation aims to provide a clear understanding of the process and the underlying mathematical principles.

Isolating the Exponential Term

The initial step in solving the equation is to isolate the exponential term. We have the equation:

$$-7 imes 3^{0.25x} = -10$$

To isolate the term $3^{0.25x}$, we divide both sides of the equation by -7:

$$\frac{-7 imes 3^{0.25x}}{-7} = \frac{-10}{-7}$$

This simplifies to:

$$3^{0.25x} = \frac{10}{7}$$

Isolating the exponential term is a crucial step because it allows us to apply logarithms effectively. By isolating the exponential part, we set the stage for using logarithms to bring down the exponent, which is our next key step. This manipulation is essential in transforming an exponential equation into a logarithmic one, which is generally easier to solve for the variable. Now that we have isolated the exponential term, we can proceed to apply logarithms to both sides of the equation.

Applying Logarithms

Having isolated the exponential term, our next step involves applying logarithms to both sides of the equation. This is a standard technique used to solve exponential equations, as it allows us to bring the exponent down using logarithmic properties. The equation we have is:

$$3^{0.25x} = \frac{10}{7}$$

To proceed, we take the logarithm of base 3 (denoted as $\log_3$) on both sides. The choice of base 3 is strategic because it simplifies the left side of the equation due to the property $\log_b(b^x) = x$. Applying the logarithm, we get:

$$\log_3(3^{0.25x}) = \log_3\left(\frac{10}{7}\right)$$

Using the logarithmic power rule, which states that $\log_b(a^c) = c \log_b(a)$, we can simplify the left side of the equation:

$$0.25x imes \log_3(3) = \log_3\left(\frac{10}{7}\right)$$

Since $\log_3(3) = 1$, the equation further simplifies to:

$$0.25x = \log_3\left(\frac{10}{7}\right)$$

Applying logarithms is a pivotal step as it transforms the exponential equation into a linear equation involving the variable x. By using the appropriate base for the logarithm, in this case, base 3, we simplify the equation significantly. The power rule of logarithms is instrumental in bringing the exponent down, making it accessible for further algebraic manipulation. Now that we have a simplified equation, we can proceed to solve for x by isolating it.

Solving for x

Now that we have the equation:

$$0.25x = \log_3\left(\frac{10}{7}\right)$$

We need to isolate x. Recall that $0.25$ is equivalent to $\frac{1}{4}$. Thus, we can rewrite the equation as:

$$\frac{1}{4}x = \log_3\left(\frac{10}{7}\right)$$

To solve for x, we multiply both sides of the equation by 4:

$$4 imes \frac{1}{4}x = 4 imes \log_3\left(\frac{10}{7}\right)$$

This simplifies to:

$$x = 4 \log_3\left(\frac{10}{7}\right)$$

Therefore, the solution to the equation $-7 imes 3^{0.25x} = -10$ is:

$$x = 4 \log_3\left(\frac{10}{7}\right)$$

Isolating the variable x involves basic algebraic manipulations, such as multiplying both sides of the equation by a constant. This step is crucial in obtaining the final solution and expressing x in terms of logarithmic functions. The solution we have found represents the exact value of x that satisfies the original equation. It is important to express the solution in its exact form whenever possible, as this avoids any rounding errors that may occur with decimal approximations.

Conclusion

In summary, we have solved the exponential equation $-7 imes 3^{0.25x} = -10$ by following a systematic approach:

1. Isolated the exponential term by dividing both sides by -7.
2. Applied logarithms (base 3) to both sides to utilize the logarithmic power rule.
3. Simplified the equation using logarithmic properties.
4. Solved for x by multiplying both sides by 4.

The solution we found is:

$$x = 4 \log_3\left(\frac{10}{7}\right)$$

This matches option (A) in the provided choices. Understanding how to solve exponential equations is a fundamental skill in mathematics, with applications ranging from simple algebraic problems to complex scientific models. The key techniques involve isolating the exponential term, applying logarithms, and using logarithmic properties to simplify and solve for the unknown variable. By mastering these techniques, one can tackle a wide range of exponential equations with confidence. The ability to solve exponential equations is not only crucial for academic pursuits but also for practical applications in various fields such as finance, engineering, and physics, where exponential models are frequently used to describe growth, decay, and other dynamic processes.

Therefore, the correct answer is (A) $x = 4 \log_3\left(\frac{10}{7}\right)$.