Solving Exponential Equations Rounding To Nearest Hundredth

by ADMIN 60 views

Introduction

In the realm of mathematics, exponential equations hold a significant place, often appearing in various scientific and engineering applications. These equations involve variables in the exponents, requiring specific techniques to solve. This article delves into the process of solving exponential equations, with a focus on rounding the solution to the nearest hundredth. We will explore the fundamental concepts, methods, and practical steps involved in tackling these equations. Consider this specific equation: 3 * 4^(x-2) = 1. Our goal is to find the value of x that satisfies this equation and then round it to the nearest hundredth. This task highlights the importance of precision in mathematical calculations and the application of rounding rules.

Understanding Exponential Equations

Exponential equations are characterized by having the variable in the exponent. The general form of an exponential equation is a^(f(x)) = b, where a is the base, f(x) is a function of x in the exponent, and b is a constant. To effectively solve exponential equations, it is crucial to grasp the properties of exponents and logarithms. Key properties include the product rule, quotient rule, power rule, and the change of base formula. These properties allow us to manipulate the equation and isolate the variable. Understanding exponential equations also involves recognizing that the exponential function is a one-to-one function, which means that if a^m = a^n, then m = n. This property is particularly useful when we can express both sides of the equation with the same base. Moreover, exponential equations are frequently encountered in real-world scenarios, such as population growth, compound interest, and radioactive decay, making their understanding and solution essential in various fields.

Methods to Solve Exponential Equations

Several methods can be employed to solve exponential equations, each suited to different scenarios. One common method involves expressing both sides of the equation with the same base. This approach leverages the one-to-one property of exponential functions. For example, if we have an equation like 2^(x+1) = 8, we can rewrite 8 as 2^3 and then equate the exponents: x + 1 = 3. Another powerful method is using logarithms. Logarithms are the inverse of exponential functions, allowing us to bring the exponent down as a coefficient. If we have an equation like 5^x = 20, we can take the logarithm of both sides: log(5^x) = log(20). Using the power rule of logarithms, we get xlog(5) = log(20)*, which can be solved for x. The choice of logarithm base (e.g., base 10 or natural logarithm) often depends on the calculator's capabilities and personal preference. Additionally, some exponential equations may require algebraic manipulation or substitution to simplify them before applying logarithmic methods. Mastering these methods is crucial for effectively solving a wide range of exponential equations and understanding their applications in various mathematical and scientific contexts.

Solving 3 * 4^(x-2) = 1 Step-by-Step

Now, let's tackle the given equation: 3 * 4^(x-2) = 1. Our aim is to isolate x and find its value. Here's a step-by-step approach:

  1. Isolate the exponential term: Divide both sides of the equation by 3: 4^(x-2) = 1/3

  2. Apply logarithms: Take the logarithm of both sides. We can use the natural logarithm (ln) or the common logarithm (log base 10). Let's use the natural logarithm: ln(4^(x-2)) = ln(1/3)

  3. Use the power rule of logarithms: Bring the exponent down as a coefficient: (x-2) * ln(4) = ln(1/3)

  4. Solve for (x-2): Divide both sides by ln(4): x - 2 = ln(1/3) / ln(4)

  5. Isolate x: Add 2 to both sides: x = (ln(1/3) / ln(4)) + 2

This step-by-step process demonstrates the systematic approach to solving exponential equations, ensuring each step logically leads to the isolation and solution of the variable.

Calculating the Value of x

To find the numerical value of x, we need to calculate the expression we derived in the previous step: x = (ln(1/3) / ln(4)) + 2. We'll use a calculator to evaluate the logarithms and perform the arithmetic operations.

  1. Calculate ln(1/3): The natural logarithm of 1/3 is approximately -1.0986.

  2. Calculate ln(4): The natural logarithm of 4 is approximately 1.3863.

  3. Divide ln(1/3) by ln(4): -1.0986 / 1.3863 ≈ -0.7925

  4. Add 2 to the result: x ≈ -0.7925 + 2 = 1.2075

Therefore, the value of x is approximately 1.2075. This calculation involves understanding logarithmic properties and using a calculator to find the numerical values, a crucial skill in solving exponential equations.

Rounding to the Nearest Hundredth

Now that we have the value of x as approximately 1.2075, we need to round it to the nearest hundredth. Rounding is a process of approximating a number to a specified decimal place. In this case, we want to round to two decimal places.

To round to the nearest hundredth, we look at the digit in the thousandths place (the third digit after the decimal). If this digit is 5 or greater, we round up the digit in the hundredths place. If it is less than 5, we leave the digit in the hundredths place as it is.

In our case, x ≈ 1.2075, the digit in the thousandths place is 7, which is greater than 5. Therefore, we round up the digit in the hundredths place (0) by one.

So, x rounded to the nearest hundredth is 1.21. This final step emphasizes the importance of precision and understanding rounding rules in mathematical calculations, providing a practical and accurate solution to the exponential equation.

Conclusion

In conclusion, solving the exponential equation 3 * 4^(x-2) = 1 required a systematic approach involving isolating the exponential term, applying logarithms, and using logarithmic properties to solve for x. We found that x is approximately 1.2075, which, when rounded to the nearest hundredth, gives us 1.21. This process highlights the importance of understanding exponential equations, logarithmic functions, and rounding rules. Exponential equations are fundamental in various fields of mathematics and science, making the ability to solve them accurately and efficiently a valuable skill. The step-by-step methodology presented here provides a clear framework for tackling similar equations and emphasizes the need for precision and attention to detail in mathematical problem-solving. The skills acquired in solving such equations extend beyond the classroom, finding application in real-world scenarios where exponential growth and decay are prevalent. Mastering these techniques ensures a solid foundation in mathematical problem-solving and fosters a deeper understanding of the principles underlying exponential relationships.