Solving 3^(2x) = 5 A Step-by-Step Guide

In the realm of mathematics, exponential equations play a crucial role, often appearing in various fields such as finance, physics, and computer science. These equations involve variables in the exponents, making them distinct from polynomial equations. One common type of exponential equation is the form a^(f(x)) = b, where 'a' and 'b' are constants, and 'f(x)' is a function of 'x'. Solving such equations requires a different approach compared to solving linear or quadratic equations. In this comprehensive guide, we will delve into the step-by-step process of solving the exponential equation 3^(2x) = 5, with a particular emphasis on rounding the answer to the nearest ten-thousandth. We will explore the underlying principles, mathematical techniques, and practical considerations involved in finding the solution. Understanding the intricacies of solving exponential equations not only enhances mathematical proficiency but also provides valuable tools for tackling real-world problems that exhibit exponential behavior. So, let's embark on this mathematical journey and unravel the solution to this intriguing equation.

Before we dive into the specifics of solving 3^(2x) = 5, it is essential to grasp the fundamental concepts of exponential equations. An exponential equation is an equation in which the variable appears in the exponent. The general form of an exponential equation is a^(f(x)) = b, where 'a' is the base, 'f(x)' is the exponent (which is a function of 'x'), and 'b' is a constant. The key characteristic of exponential equations is that the variable is part of the exponent, which means that the rate of change is not constant but rather increases or decreases exponentially. To solve exponential equations, we often employ logarithms, which are the inverse functions of exponential functions. Logarithms allow us to "bring down" the variable from the exponent, making the equation solvable. There are two main types of logarithms: common logarithms (base 10) and natural logarithms (base e). The choice of logarithm base depends on the specific equation, but the natural logarithm is often preferred due to its mathematical properties and convenience in calculus and other advanced fields. Understanding the relationship between exponential functions and logarithms is crucial for mastering the art of solving exponential equations. The ability to manipulate logarithmic expressions and apply logarithmic properties is key to isolating the variable and finding the solution.

Now, let's proceed with the step-by-step solution of the given exponential equation, 3^(2x) = 5. This process involves a series of carefully executed steps, each building upon the previous one, to isolate the variable 'x' and determine its value. By meticulously following these steps, we can gain a deeper understanding of the solution process and enhance our problem-solving skills in the context of exponential equations.

Step 1: Applying Logarithms

The first crucial step in solving the equation 3^(2x) = 5 is to apply logarithms to both sides of the equation. The logarithm function allows us to bring the exponent down, making it easier to isolate the variable. We can choose either the common logarithm (base 10) or the natural logarithm (base e). In this case, let's use the natural logarithm (ln) for its mathematical convenience. Applying the natural logarithm to both sides of the equation, we get:

ln(3^(2x)) = ln(5)

The application of logarithms is a fundamental step in solving exponential equations. It leverages the inverse relationship between exponential functions and logarithmic functions, enabling us to transform the equation into a more manageable form.

Step 2: Using Logarithmic Properties

Next, we utilize the logarithmic property that states ln(a^b) = b * ln(a). This property is essential for simplifying the equation and bringing the variable 'x' out of the exponent. Applying this property to the left side of the equation, we have:

2x * ln(3) = ln(5)

The logarithmic property employed here is a cornerstone of logarithmic manipulation. It allows us to transform an expression involving a logarithm of a power into a product of the exponent and the logarithm of the base. This transformation is crucial for isolating the variable in exponential equations.

Step 3: Isolating the Variable

Now, we need to isolate the variable 'x'. To do this, we divide both sides of the equation by 2 * ln(3):

x = ln(5) / (2 * ln(3))

This step involves algebraic manipulation to separate 'x' from the other terms in the equation. By dividing both sides by the coefficient of 'x', we isolate the variable and express it in terms of logarithmic expressions.

Step 4: Calculating the Value

Using a calculator, we compute the value of the expression ln(5) / (2 * ln(3)). The natural logarithm of 5 is approximately 1.6094, and the natural logarithm of 3 is approximately 1.0986. Therefore, we have:

x ≈ 1.6094 / (2 * 1.0986)

x ≈ 1.6094 / 2.1972

x ≈ 0.732478

This step involves the use of a calculator to evaluate the logarithmic expressions and obtain a numerical approximation for 'x'. The accuracy of the result depends on the precision of the calculator and the number of decimal places used in the calculation.

Step 5: Rounding to the Nearest Ten-Thousandth

Finally, we round the value of 'x' to the nearest ten-thousandth, which is four decimal places:

x ≈ 0.7325

Rounding is a crucial step in many mathematical problems, especially when dealing with approximations and decimal numbers. Rounding to the nearest ten-thousandth ensures that the answer is accurate to four decimal places, which is often sufficient for practical applications.

Therefore, the solution to the equation 3^(2x) = 5, rounded to the nearest ten-thousandth, is approximately x ≈ 0.7325. This final answer represents the value of 'x' that satisfies the given exponential equation. The process of solving this equation has demonstrated the application of logarithms, logarithmic properties, and algebraic manipulation to isolate the variable and find its value. This step-by-step approach can be applied to solving other exponential equations as well.

While using logarithms is the most common method for solving exponential equations, there are alternative approaches that can be employed depending on the specific equation. These methods provide different perspectives and can be useful in certain situations.

Method 1: Using Common Logarithms (Base 10)

Instead of using natural logarithms (base e), we can use common logarithms (base 10). The process is similar, but we use the common logarithm function (log) instead of the natural logarithm function (ln). Applying the common logarithm to both sides of the equation 3^(2x) = 5, we get:

log(3^(2x)) = log(5)

Using the logarithmic property log(a^b) = b * log(a), we have:

2x * log(3) = log(5)

Isolating 'x', we get:

x = log(5) / (2 * log(3))

Using a calculator, we find that log(5) ≈ 0.6990 and log(3) ≈ 0.4771. Therefore:

x ≈ 0.6990 / (2 * 0.4771)

x ≈ 0.6990 / 0.9542

x ≈ 0.7325

Rounding to the nearest ten-thousandth, we get x ≈ 0.7325, which is the same solution as before. This method demonstrates that using common logarithms yields the same result as using natural logarithms.

Method 2: Graphical Method

Another way to solve exponential equations is by using a graphical method. This involves plotting the graphs of both sides of the equation and finding the point of intersection. For the equation 3^(2x) = 5, we can plot the graphs of y = 3^(2x) and y = 5. The x-coordinate of the point where the two graphs intersect is the solution to the equation. Using a graphing calculator or software, we can plot these graphs and find the intersection point. The x-coordinate of the intersection point will be approximately 0.7325, which is consistent with our previous solution. The graphical method provides a visual representation of the solution and can be useful for understanding the behavior of exponential functions.

Method 3: Numerical Methods

For more complex exponential equations that cannot be solved algebraically, numerical methods can be employed. These methods involve iterative algorithms that approximate the solution to a desired degree of accuracy. Examples of numerical methods include the Newton-Raphson method and the bisection method. These methods are often used in computer software and calculators to solve equations that do not have closed-form solutions. While numerical methods are beyond the scope of this article, they represent a powerful tool for solving a wide range of mathematical problems.

When solving exponential equations, there are several common mistakes that students often make. Being aware of these mistakes and taking steps to avoid them can improve accuracy and problem-solving skills.

Mistake 1: Incorrectly Applying Logarithmic Properties

One of the most common mistakes is misapplying logarithmic properties. For example, students may incorrectly assume that ln(a + b) = ln(a) + ln(b), which is not true. The correct logarithmic properties are:

• ln(a * b) = ln(a) + ln(b)

• ln(a / b) = ln(a) - ln(b)

• ln(a^b) = b * ln(a)

It is crucial to understand and apply these properties correctly to avoid errors.

Mistake 2: Not Applying Logarithms to Both Sides

Another common mistake is applying logarithms to only one side of the equation. To maintain equality, logarithms must be applied to both sides of the equation. For example, if we have 3^(2x) = 5, we must apply the logarithm to both sides:

ln(3^(2x)) = ln(5)

Failing to do so will lead to an incorrect solution.

Mistake 3: Incorrect Order of Operations

Following the correct order of operations is essential in mathematics. When solving exponential equations, it is important to perform operations in the correct sequence. For example, in the equation x = ln(5) / (2 * ln(3)), we must first calculate 2 * ln(3) and then divide ln(5) by the result. Incorrect order of operations can lead to significant errors.

Mistake 4: Rounding Too Early

Rounding intermediate results too early can affect the accuracy of the final answer. It is best to perform calculations with as many decimal places as possible and round only at the end. For example, if we round ln(5) and ln(3) to fewer decimal places before calculating x, the final result may be less accurate.

Mistake 5: Not Checking the Solution

It is always a good practice to check the solution by substituting it back into the original equation. This helps to verify that the solution is correct and that no mistakes were made during the solving process. For example, if we found x ≈ 0.7325, we can substitute it back into the equation 3^(2x) = 5 to check:

3^(2 * 0.7325) ≈ 3^(1.465) ≈ 5

If the equation holds true, then the solution is correct.

Exponential equations are not just abstract mathematical concepts; they have numerous real-world applications in various fields. Understanding exponential equations can help us model and analyze phenomena in finance, science, and technology.

Finance

In finance, exponential equations are used to model compound interest. The formula for compound interest is:

A = P(1 + r/n)^(nt)

where:

• A is the future value of the investment

• P is the principal amount

• r is the annual interest rate

• n is the number of times interest is compounded per year

• t is the number of years

This equation is an exponential equation because the variable t is in the exponent. Solving exponential equations allows us to determine the time it takes for an investment to reach a certain value or the interest rate required to achieve a financial goal.

Science

In science, exponential equations are used to model population growth, radioactive decay, and chemical reactions. For example, the population growth of a species can be modeled by the exponential equation:

N(t) = N_0 * e^(kt)

where:

• N(t) is the population at time t

• N_0 is the initial population

• k is the growth rate

• e is the base of the natural logarithm

Similarly, radioactive decay can be modeled by the exponential equation:

N(t) = N_0 * e^(-λt)

where:

• N(t) is the amount of radioactive substance at time t

• N_0 is the initial amount

• λ is the decay constant

Technology

In technology, exponential equations are used in various applications, such as computer science and signal processing. For example, the growth of computing power, as described by Moore's Law, is exponential. Moore's Law states that the number of transistors on a microchip doubles approximately every two years, which can be modeled by an exponential equation.

In this comprehensive guide, we have explored the step-by-step solution of the exponential equation 3^(2x) = 5, rounding the answer to the nearest ten-thousandth. We have discussed the underlying principles of exponential equations, the application of logarithms, alternative methods for solving exponential equations, common mistakes to avoid, and real-world applications of exponential equations. By mastering the techniques presented in this guide, you can enhance your mathematical proficiency and problem-solving skills in the context of exponential equations. The ability to solve exponential equations is a valuable skill that can be applied to various fields, including finance, science, and technology. Whether you are a student, a professional, or simply someone interested in mathematics, understanding exponential equations will undoubtedly broaden your mathematical horizons and open doors to new possibilities.

To solidify your understanding of solving exponential equations, try solving the following problems:

1. Solve 2^(3x) = 7, rounding to the nearest ten-thousandth.
2. Solve 5^(x - 1) = 12, rounding to the nearest ten-thousandth.
3. Solve 4^(2x + 1) = 9, rounding to the nearest ten-thousandth.

By practicing these problems, you will reinforce your skills and gain confidence in solving exponential equations.