Solving Exponential Equations E^m = 28 A Step-by-Step Guide

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When faced with the exponential equation em=28e^m = 28, our goal is to isolate the variable m. This requires understanding the relationship between exponential and logarithmic functions, specifically the natural logarithm. In this article, we will delve into a step-by-step solution, providing a clear explanation for each stage, and discuss why the natural logarithm is the key to unlocking this problem. Furthermore, we will explore the broader applications of solving exponential equations, emphasizing their significance in various fields, including finance, science, and engineering.

The core concept in solving equations of this type lies in the inverse relationship between exponential functions and logarithms. The natural exponential function, denoted as exe^x, has a base of e, which is approximately 2.71828. The inverse of this function is the natural logarithm, denoted as ln(x). By applying the natural logarithm to both sides of the equation, we can effectively undo the exponential operation and isolate the variable. This technique is not only crucial for solving this specific problem but also forms the foundation for tackling more complex exponential equations. The natural logarithm essentially asks the question: "To what power must we raise e to obtain a given value?" In the context of our equation, it helps us find the exponent m that satisfies the condition em=28e^m = 28.

To solve the equation em=28e^m = 28, we take the natural logarithm (ln) of both sides. This is a crucial step because it allows us to use the property of logarithms that states ln(aba^b) = b * ln(a). Applying this to our equation, we get:

ln(eme^m) = ln(28)

Using the logarithmic property, the left side simplifies to:

m * ln(e) = ln(28)

Since ln(e) is equal to 1 (because e raised to the power of 1 is e), the equation further simplifies to:

m = ln(28)

Now, we need to find the value of ln(28). This can be done using a calculator equipped with a natural logarithm function. The natural logarithm of 28 is approximately 3.3322. Therefore, the solution to the equation em=28e^m = 28 is:

m ≈ 3.3322

This result means that e raised to the power of approximately 3.3322 equals 28. Rounding to the nearest ten-thousandth, as requested, confirms that option D, 3.3322, is the correct answer. It's essential to note that the precision of the answer depends on the precision of the calculator used and the rounding rules applied. In many practical applications, understanding the level of accuracy required is critical. For instance, in financial calculations, even small differences in exponents can lead to significant variations in results over time.

To solve the equation em=28e^m = 28 and understand the underlying mathematical principles, we will follow a detailed, step-by-step approach. This process not only provides the solution but also clarifies the logic behind each step, enhancing comprehension and problem-solving skills. The use of the natural logarithm is paramount in this process, and we will explain its role and significance in detail.

1. Start with the Equation:

We are given the equation:

em=28e^m = 28

Here, e is the base of the natural exponential function, and m is the exponent we need to find. The number 28 is the result of raising e to the power of m. To isolate m, we need to apply the inverse operation of exponentiation, which is taking the logarithm.

2. Apply the Natural Logarithm to Both Sides:

The key to solving this type of equation is to take the natural logarithm (ln) of both sides. The natural logarithm is the logarithm to the base e. This is crucial because the natural logarithm and the exponential function with base e are inverse functions. Applying the natural logarithm to both sides, we get:

ln(eme^m) = ln(28)

This step is justified because if two quantities are equal, their logarithms (to the same base) are also equal. The introduction of the natural logarithm allows us to manipulate the equation in a way that isolates the variable m.

3. Use the Logarithmic Power Rule:

The next step involves using a fundamental property of logarithms, the power rule. The power rule states that for any positive number a, any exponent b, and any base c (where c > 0 and c ≠ 1):

ln(aba^b) = b * ln(a)

Applying this rule to the left side of our equation, ln(eme^m), we get:

m * ln(e) = ln(28)

This transformation is essential because it brings the exponent m down from the exponential position to a multiplicative position, making it easier to isolate.

4. Simplify Using ln(e) = 1:

One of the most important properties of the natural logarithm is that the natural logarithm of e is equal to 1. This is because e raised to the power of 1 is e. Therefore:

ln(e) = 1

Substituting this into our equation, we have:

m * 1 = ln(28)

Which simplifies to:

m = ln(28)

This simplification is crucial as it directly relates m to the natural logarithm of 28, which can be computed numerically.

5. Calculate ln(28) Using a Calculator:

To find the numerical value of m, we need to calculate ln(28). This is typically done using a scientific calculator that has a natural logarithm function. Inputting 28 into the natural logarithm function yields:

ln(28) ≈ 3.332204510175204

This value is an approximation because ln(28) is an irrational number, meaning it has a non-repeating, non-terminating decimal expansion. Calculators provide an approximation up to a certain number of decimal places.

6. Round the Answer to the Nearest Ten-Thousandth:

The problem specifies that the answer should be rounded to the nearest ten-thousandth. The ten-thousandth place is the fourth digit after the decimal point. Looking at our calculator result, we have:

  1. 332204510175204

The digit in the ten-thousandth place is 2. The digit to the right of it is 0, which is less than 5, so we round down. This means we keep the 2 as it is and truncate the remaining digits. Therefore, the rounded value is:

m ≈ 3.3322

This is the final solution to the equation, rounded to the specified precision. It's crucial to pay attention to rounding instructions, as different levels of precision may be required in different contexts.

7. Verify the Solution (Optional):

To ensure our solution is correct, we can substitute the rounded value of m back into the original equation and check if it holds true:

e3.3322e^{3.3322} ≈ 28

Using a calculator, we find that:

e3.3322e^{3.3322} ≈ 27.9998

This is very close to 28, and the slight difference is due to rounding errors. The solution is therefore verified to be correct.

Through these detailed steps, we have shown how to solve the equation em=28e^m = 28 using the natural logarithm. Each step is logically connected, and understanding these steps provides a solid foundation for solving similar exponential equations. The key takeaways are the application of the natural logarithm, the use of the power rule of logarithms, and the simplification using ln(e) = 1.

The natural logarithm, denoted as ln(x), plays a pivotal role in solving exponential equations, especially those involving the base e. Understanding why it is the key to unlocking these problems requires a clear grasp of its definition and properties. The natural logarithm is the inverse function of the natural exponential function, exe^x, and this inverse relationship is what makes it so effective in isolating variables within exponents.

Definition and Inverse Relationship

The natural logarithm of a number x is the power to which e must be raised to equal x. Mathematically, if ey=xe^y = x, then ln(x) = y. This inverse relationship means that applying the natural logarithm to an exponential function with base e effectively "undoes" the exponentiation. In other words:

ln(exe^x) = x

and

eln(x)e^{ln(x)} = x

This fundamental property is the cornerstone of solving equations like em=28e^m = 28. By taking the natural logarithm of both sides, we can bring the exponent down and isolate the variable. Without the natural logarithm, manipulating exponential equations would be significantly more complex.

Properties of Natural Logarithms

Several properties of natural logarithms make them indispensable in mathematical manipulations. These properties stem from the properties of exponents and provide a set of tools for simplifying and solving equations. The most relevant properties for solving exponential equations include:

  1. Product Rule: ln(xy) = ln(x) + ln(y)
  2. Quotient Rule: ln(x/y) = ln(x) - ln(y)
  3. Power Rule: ln(xpx^p) = p * ln(x)

The power rule is particularly crucial in solving exponential equations. As demonstrated in the step-by-step solution above, applying the power rule to ln(eme^m) allows us to rewrite the expression as m * ln(e). Since ln(e) = 1, this simplifies to m, effectively isolating the variable.

Significance of the Base e

The base e, an irrational number approximately equal to 2.71828, is central to many areas of mathematics and science. It arises naturally in calculus, compound interest, exponential growth and decay models, and numerous other applications. The natural logarithm is tailored to this base, making it the most natural and efficient choice for dealing with exponential functions involving e.

In contrast, logarithms with other bases, such as base 10 (common logarithm), could be used, but they would introduce an additional step of conversion. The natural logarithm directly aligns with the exponential function with base e, eliminating the need for such conversions and streamlining the solution process.

Applications Beyond Basic Equations

The utility of the natural logarithm extends far beyond solving simple equations like em=28e^m = 28. It is an essential tool in various advanced mathematical and scientific contexts:

  • Calculus: The derivative of exe^x is itself, and the derivative of ln(x) is 1/x. These properties make the natural exponential and logarithmic functions fundamental in differential and integral calculus.
  • Differential Equations: Many physical processes, such as radioactive decay and population growth, are modeled by differential equations involving exponential functions. Solving these equations often requires the use of natural logarithms.
  • Finance: Compound interest and continuous compounding formulas rely heavily on the natural exponential function and its inverse, the natural logarithm.
  • Statistics: The normal distribution, a cornerstone of statistical analysis, is defined using the exponential function with base e.
  • Physics and Engineering: Exponential functions and natural logarithms appear in models of electrical circuits, heat transfer, fluid dynamics, and quantum mechanics.

In each of these areas, the natural logarithm provides a means to simplify complex expressions, solve equations, and gain insights into the behavior of various systems. Its role as the inverse of the natural exponential function makes it an indispensable tool for mathematicians, scientists, and engineers.

In conclusion, the natural logarithm is the key to solving equations involving the natural exponential function due to its inverse relationship with e^x and its unique properties. Its significance extends far beyond basic algebra, making it a fundamental concept in numerous scientific and mathematical disciplines. By understanding the natural logarithm, one can effectively tackle a wide range of problems, from simple exponential equations to complex models of natural phenomena.

When solving exponential equations, particularly those involving the natural logarithm, it is crucial to avoid common mistakes that can lead to incorrect answers. These mistakes often stem from a misunderstanding of the properties of logarithms or from algebraic errors in manipulating equations. Recognizing and avoiding these pitfalls can significantly improve accuracy and problem-solving efficiency.

1. Incorrectly Applying Logarithmic Properties

One of the most frequent errors is misapplying the properties of logarithms. While the logarithmic properties provide powerful tools for simplifying equations, they must be used correctly. Common mistakes include:

  • Misusing the Product Rule: The product rule states that ln(xy) = ln(x) + ln(y). A common mistake is to assume that ln(x + y) = ln(x) + ln(y), which is incorrect. The logarithm of a sum is not equal to the sum of the logarithms.
  • Misusing the Quotient Rule: Similarly, the quotient rule states that ln(x/y) = ln(x) - ln(y). An incorrect application might involve assuming ln(x - y) = ln(x) - ln(y), which is also false.
  • Misusing the Power Rule: The power rule, ln(xpx^p) = p * ln(x), is often misused when dealing with complex expressions. It is essential to ensure that the exponent applies to the entire argument of the logarithm before applying the rule.

To avoid these mistakes, it is vital to memorize and understand the correct logarithmic properties and practice applying them in various contexts. Always double-check each step to ensure that the properties are used accurately.

2. Forgetting to Apply the Logarithm to Both Sides

When solving an exponential equation, it is essential to apply the logarithm (natural or otherwise) to both sides of the equation. Applying it to only one side will disrupt the equality and lead to an incorrect solution. For example, in the equation em=28e^m = 28, taking the natural logarithm of only the left side would yield ln(eme^m) = m, but without taking the natural logarithm of the right side, we cannot isolate m and find its value.

To avoid this mistake, always remember that any operation performed on one side of an equation must also be performed on the other side to maintain balance. This fundamental principle of algebra applies to all types of equations, including exponential and logarithmic ones.

3. Incorrectly Simplifying ln(e)

The natural logarithm of e, ln(e), is equal to 1. This simplification is a crucial step in solving equations involving exe^x. A common mistake is to overlook this simplification or to incorrectly assign a different value to ln(e). Remembering that ln(e) = 1 is essential for accurately solving exponential equations.

4. Rounding Errors

Rounding errors can significantly affect the accuracy of the final answer, especially when dealing with irrational numbers like ln(28). It is crucial to follow the instructions regarding rounding and to avoid premature rounding. Rounding intermediate results can lead to accumulated errors that deviate the final answer from the correct value.

In the case of em=28e^m = 28, calculating ln(28) using a calculator gives approximately 3.332204510175204. If the instructions specify rounding to the nearest ten-thousandth, the correct answer is 3.3322. Rounding to 3.332 or 3.33 can introduce significant errors, particularly in applications where precision is critical.

To minimize rounding errors, perform calculations with as many decimal places as possible and round only the final answer to the specified precision.

5. Misunderstanding the Domain of Logarithms

Logarithms are only defined for positive numbers. The natural logarithm, ln(x), is only defined for x > 0. Attempting to take the logarithm of a non-positive number (zero or a negative number) will result in an undefined value.

In solving logarithmic equations, it is crucial to check the solutions to ensure they are within the domain of the logarithmic functions involved. Extraneous solutions, which are solutions that satisfy the transformed equation but not the original equation, can arise from this issue.

6. Neglecting the Inverse Relationship

The inverse relationship between the exponential function exe^x and the natural logarithm ln(x) is fundamental. Forgetting this relationship can lead to confusion and errors. Remember that ln(exe^x) = x and eln(x)e^{ln(x)} = x. These identities are the key to simplifying and solving exponential and logarithmic equations.

7. Algebraic Errors

Simple algebraic errors, such as incorrect addition, subtraction, multiplication, or division, can also lead to incorrect solutions. Always double-check each step of the algebraic manipulation to ensure accuracy.

In conclusion, avoiding common mistakes when solving exponential equations requires a thorough understanding of logarithmic properties, careful algebraic manipulation, adherence to rounding rules, and an awareness of the domain of logarithmic functions. By recognizing these pitfalls and practicing accurate problem-solving techniques, one can confidently tackle a wide range of exponential and logarithmic equations.

Exponential equations, like the one we solved, em=28e^m = 28, might seem abstract, but they are powerful tools with numerous real-world applications. These equations model phenomena in various fields, including finance, biology, physics, and engineering. Understanding the principles behind solving exponential equations allows us to analyze and predict outcomes in these diverse areas. Let's explore some specific examples of how exponential equations are used in practice.

1. Finance: Compound Interest

One of the most common applications of exponential equations is in calculating compound interest. The formula for continuous compounding is given by:

A=P∗ertA = P * e^{rt}

Where:

  • A is the amount of money accumulated after t years, including interest.
  • P is the principal amount (the initial investment).
  • r is the annual interest rate (as a decimal).
  • t is the number of years the money is invested.
  • e is the base of the natural logarithm (approximately 2.71828).

In this formula, the exponential term erte^{rt} plays a crucial role in determining the growth of the investment over time. If we want to find out how long it takes for an investment to reach a specific amount, we would need to solve an exponential equation. For instance, if an initial investment of $1000 is made at an annual interest rate of 5%, we can calculate how long it will take for the investment to double by solving the equation:

2000=1000∗e0.05t2000 = 1000 * e^{0.05t}

Dividing both sides by 1000, we get:

2 = e0.05te^{0.05t}

Taking the natural logarithm of both sides:

ln(2) = 0.05t

Solving for t:

t=ln(2)0.05t = \frac{ln(2)}{0.05} ≈ 13.86 years

This example demonstrates how solving an exponential equation allows us to determine the time required for an investment to reach a specific target.

2. Biology: Population Growth and Decay

Exponential equations are fundamental in modeling population growth and decay. In ideal conditions, a population of living organisms (such as bacteria) grows exponentially. The population N at time t can be modeled by the equation:

N(t)=N0∗ektN(t) = N_0 * e^{kt}

Where:

  • N(t) is the population at time t.
  • N0N_0 is the initial population.
  • k is the growth rate constant (positive for growth, negative for decay).

If we know the initial population and the growth rate, we can use this equation to predict the population at any future time. Conversely, if we have population data at different times, we can solve for the growth rate constant k. For example, if a population of bacteria doubles in 2 hours, we can find the growth rate constant k by solving:

2N0N_0 = N0∗e2kN_0 * e^{2k}

Dividing both sides by N0N_0:

2 = e2ke^{2k}

Taking the natural logarithm of both sides:

ln(2) = 2k

Solving for k:

k=ln(2)2k = \frac{ln(2)}{2} ≈ 0.3466

This growth rate constant can then be used to predict the population at other times.

Exponential decay is also modeled using similar equations, but with a negative growth rate constant. Radioactive decay, for instance, follows an exponential decay model, where the amount of a radioactive substance decreases over time according to the equation:

A(t)=A0∗e−λtA(t) = A_0 * e^{-λt}

Where:

  • A(t) is the amount of the substance at time t.
  • A0A_0 is the initial amount of the substance.
  • λ (lambda) is the decay constant.

Solving for t in this equation allows us to determine the half-life of a radioactive substance, which is the time it takes for half of the substance to decay.

3. Physics: Radioactive Decay

In physics, radioactive decay is a prime example of an exponential process. The number of radioactive nuclei N in a sample decreases exponentially with time t, as described by the equation:

N(t)=N0∗e−λtN(t) = N_0 * e^{-λt}

Where:

  • N(t) is the number of radioactive nuclei at time t.
  • N0N_0 is the initial number of radioactive nuclei.
  • λ (lambda) is the decay constant, which is characteristic of the radioactive substance.

The half-life (t1/2t_{1/2}) of a radioactive substance is the time required for half of the initial nuclei to decay. It is related to the decay constant by the equation:

t1/2=ln(2)λt_{1/2} = \frac{ln(2)}{λ}

Solving exponential equations allows physicists to determine the age of ancient artifacts using carbon dating, a technique based on the radioactive decay of carbon-14. By measuring the remaining amount of carbon-14 in a sample and knowing its half-life, the age of the sample can be estimated.

4. Engineering: Electrical Circuits

Exponential equations are also used in electrical engineering to analyze circuits containing resistors and capacitors (RC circuits) or resistors and inductors (RL circuits). The voltage or current in these circuits changes exponentially over time as the capacitor charges or discharges, or as the inductor's current increases or decreases.

For example, in an RC circuit, the voltage V(t) across the capacitor as it charges can be described by the equation:

V(t)=V0∗(1−e−t/RC)V(t) = V_0 * (1 - e^{-t/RC})

Where:

  • V(t)V(t) is the voltage across the capacitor at time t.
  • V0V_0 is the maximum voltage.
  • R is the resistance.
  • C is the capacitance.

Solving for t in this equation allows engineers to determine the time it takes for the capacitor to charge to a certain voltage level. This is crucial in designing timing circuits and other electronic applications.

In summary, exponential equations are not just abstract mathematical constructs; they are powerful tools with wide-ranging applications in various fields. From finance to biology, physics, and engineering, exponential equations allow us to model and analyze growth, decay, and other dynamic processes. Understanding how to solve these equations is essential for making predictions, designing systems, and gaining insights into the world around us.

In conclusion, solving the equation em=28e^m = 28 exemplifies the importance of understanding exponential functions and their inverse relationship with natural logarithms. The detailed, step-by-step solution demonstrated how to apply the natural logarithm to both sides of the equation, utilize the power rule of logarithms, and simplify using ln(e) = 1 to isolate the variable m. The correct answer, rounded to the nearest ten-thousandth, is 3.3322.

This exercise also highlighted the crucial role of the natural logarithm as the key to solving equations involving the base e. The properties of natural logarithms, particularly the power rule, allow us to manipulate exponential expressions and find solutions efficiently. Furthermore, understanding common mistakes, such as misapplying logarithmic properties or neglecting to apply the logarithm to both sides, is essential for accurate problem-solving.

The real-world applications discussed underscore the significance of exponential equations in diverse fields, including finance, biology, physics, and engineering. From calculating compound interest and modeling population growth to analyzing radioactive decay and electrical circuits, exponential equations provide valuable tools for understanding and predicting dynamic processes. The ability to solve these equations empowers us to make informed decisions and design effective systems.

By mastering the techniques for solving exponential equations and appreciating their practical applications, individuals can enhance their mathematical proficiency and problem-solving capabilities. This knowledge is not only valuable in academic pursuits but also in various professional and personal contexts where quantitative analysis is essential. The equation em=28e^m = 28 serves as a gateway to a deeper understanding of exponential functions and their profound impact on the world around us.