Solving E^x-3=2 A Step-by-Step Guide With Rounding To Nearest Thousandth

In the realm of mathematics, exponential equations play a crucial role, particularly in modeling various real-world phenomena such as population growth, radioactive decay, and compound interest. These equations involve variables in the exponents, making them distinct from polynomial equations. Successfully navigating exponential equations requires a solid understanding of exponential functions and logarithms. This guide aims to provide a comprehensive walkthrough on how to solve exponential equations, focusing on the specific example of ${ e^x - 3 = 2 }$. We will delve into the necessary steps and concepts, ensuring a clear and concise approach to finding solutions and rounding them to the nearest thousandth.

At its core, solving an exponential equation involves isolating the exponential term and then utilizing logarithms to bring the variable out of the exponent. The process involves algebraic manipulation and the application of logarithmic properties. Logarithms, the inverse functions of exponentials, are indispensable tools in this context. The natural logarithm, denoted as ${ \ln }$, is particularly useful when dealing with the base ${ e }$, the natural exponential base. Understanding these principles is essential for tackling a wide range of exponential equations.

Before we dive into the specific example, let's lay a conceptual foundation. An exponential equation generally takes the form ${ a^x = b }$, where ${ a }$ is the base, ${ x }$ is the exponent, and ${ b }$ is the result. To solve for ${ x }$, we apply logarithms to both sides of the equation. The choice of the logarithmic base often depends on the base of the exponential term. If the base is ${ e }$, the natural logarithm is the most convenient choice. If the base is 10, the common logarithm is preferred. By strategically applying logarithms, we can simplify the equation and isolate the variable.

Understanding the properties of logarithms is paramount. The key property we'll use is the power rule, which states that ${ \log_b(a^c) = c \log_b(a) }$. This property allows us to bring the exponent down as a coefficient, thereby transforming the exponential equation into a more manageable linear equation. Additionally, we'll rely on the property that ${ \log_b(b) = 1 }$, which simplifies expressions when the base of the logarithm matches the base of the exponential term.

Step-by-Step Solution of ${ e^x - 3 = 2 }$

Now, let's apply these concepts to solve the given equation ${ e^x - 3 = 2 }$. We will meticulously walk through each step, ensuring clarity and precision.

Step 1: Isolate the Exponential Term

The first crucial step in solving any exponential equation is to isolate the exponential term on one side of the equation. In our case, the exponential term is ${ e^x }$. To isolate it, we need to eliminate the constant term, which is -3. We achieve this by adding 3 to both sides of the equation:

${ e^x - 3 + 3 = 2 + 3 }$

This simplifies to:

${ e^x = 5 }$

Step 2: Apply the Natural Logarithm

With the exponential term isolated, we now apply the natural logarithm (${ \ln }$) to both sides of the equation. The natural logarithm is the logarithm to the base ${ e }$, and it is the ideal choice for equations involving the natural exponential function. Applying the natural logarithm, we get:

${ \ln(e^x) = \ln(5) }$

Step 3: Use the Power Rule of Logarithms

Here, we invoke the power rule of logarithms, which states that ${ \ln(a^b) = b \ln(a) }$. Applying this rule to the left side of our equation, we have:

${ x \ln(e) = \ln(5) }$

Since ${ \ln(e) = 1 }$, the equation simplifies to:

${ x = \ln(5) }$

Step 4: Calculate the Natural Logarithm of 5

To find the numerical value of ${ x }$, we need to calculate the natural logarithm of 5. This can be done using a calculator with a natural logarithm function. We find that:

${ \ln(5) \approx 1.60943791243 }$

Step 5: Round to the Nearest Thousandth

The final step is to round our answer to the nearest thousandth. The thousandth place is the third digit after the decimal point. Looking at our value, 1.60943791243, the digit in the thousandth place is 9, and the digit to its right is 4. Since 4 is less than 5, we round down, keeping the 9 as it is. Thus, we round 1.60943791243 to 1.609.

Therefore, the solution to the equation ${ e^x - 3 = 2 }$, rounded to the nearest thousandth, is:

${ x \approx 1.609 }$

Further Examples and Practice Problems

To solidify your understanding, let's explore additional examples and practice problems involving exponential equations.

Example 1: Solve ${ 2^{x+1} = 16 }$

Step 1: Express both sides with the same base

In this case, we can express 16 as a power of 2: ${ 16 = 2^4 }$. So the equation becomes:

${ 2^{x+1} = 2^4 }$

Step 2: Equate the exponents

Since the bases are equal, we can equate the exponents:

${ x + 1 = 4 }$

Step 3: Solve for x

Subtract 1 from both sides:

${ x = 4 - 1 }$

${ x = 3 }$

Example 2: Solve ${ 3^x = 27 }$

Step 1: Express both sides with the same base

We can express 27 as a power of 3: ${ 27 = 3^3 }$. So the equation becomes:

${ 3^x = 3^3 }$

Step 2: Equate the exponents

Since the bases are equal, we can equate the exponents:

${ x = 3 }$

Example 3: Solve ${ 5^{2x-1} = 125 }$

Step 1: Express both sides with the same base

We can express 125 as a power of 5: ${ 125 = 5^3 }$. So the equation becomes:

${ 5^{2x-1} = 5^3 }$

Step 2: Equate the exponents

Since the bases are equal, we can equate the exponents:

${ 2x - 1 = 3 }$

Step 3: Solve for x

Add 1 to both sides:

${ 2x = 4 }$

Divide by 2:

${ x = 2 }$

Example 4: Solve ${ e^{2x} = 7 }$

Step 1: Apply the Natural Logarithm

Apply ${ \ln }$ to both sides:

${ \ln(e^{2x}) = \ln(7) }$

Step 2: Use the Power Rule of Logarithms

${ 2x \ln(e) = \ln(7) }$

Since ${ \ln(e) = 1 }$, we have:

${ 2x = \ln(7) }$

Step 3: Solve for x

Divide by 2:

${ x = \frac{\ln(7)}{2} }$

Step 4: Calculate and Round

${ x \approx \frac{1.94591014906}{2} \approx 0.973 }$

Practice Problems

1. Solve ${ 4^{x-1} = 64 }$
2. Solve ${ e^{3x} = 10 }$ (Round to the nearest thousandth)
3. Solve ${ 7^{x+2} = 343 }$
4. Solve ${ 2^{5x} = 32 }$
5. Solve ${ e^{x/2} = 4 }$ (Round to the nearest thousandth)

Common Pitfalls and How to Avoid Them

Solving exponential equations can present certain challenges if not approached carefully. Here are some common pitfalls and strategies to avoid them:

1. Incorrectly Applying Logarithms: Ensure that you apply the logarithm to the entire side of the equation, not just individual terms. For example, in the equation ${ e^x - 3 = 2 }$, you should isolate ${ e^x }$ first before applying the logarithm.

2. Forgetting the Power Rule: The power rule of logarithms is crucial for bringing the exponent down as a coefficient. Neglecting this rule can lead to significant errors. Always remember that ${ \ln(a^b) = b \ln(a) }$.

3. Rounding Errors: When rounding to the nearest thousandth, make sure to look at the digit in the ten-thousandth place to determine whether to round up or down. Rounding too early in the process can also introduce errors, so it’s best to wait until the final step.

4. Misunderstanding Logarithmic Properties: A thorough understanding of logarithmic properties is essential. For instance, remember that ${ \ln(e) = 1 }$ and ${ \log_b(b) = 1 }$. Misapplying these properties can lead to incorrect solutions.

5. Not Expressing Numbers with the Same Base: When possible, try to express both sides of the equation with the same base. This simplifies the equation and allows you to equate the exponents directly. For example, when solving ${ 2^{x+1} = 16 }$, recognize that ${ 16 = 2^4 }$.

Real-World Applications of Exponential Equations

Exponential equations are not just abstract mathematical concepts; they have numerous applications in the real world. Here are a few examples:

1. Population Growth: Exponential functions are used to model population growth. The equation ${ P(t) = P_0 e^{kt} }$ represents the population ${ P(t) }$ at time ${ t }$, where ${ P_0 }$ is the initial population and ${ k }$ is the growth rate. Solving exponential equations helps predict future population sizes.

2. Radioactive Decay: Radioactive decay follows an exponential decay model. The amount of a radioactive substance remaining after time ${ t }$ is given by ${ N(t) = N_0 e^{-\lambda t} }$, where ${ N_0 }$ is the initial amount and ${ \lambda }$ is the decay constant. Exponential equations help determine the half-life of radioactive materials.

3. Compound Interest: Compound interest is another area where exponential equations are used. The amount ${ A }$ after ${ t }$ years with principal ${ P }$, annual interest rate ${ r }$, and ${ n }$ compounding periods per year is given by ${ A = P(1 + \frac{r}{n})^{nt} }$. Solving exponential equations helps calculate the time needed for an investment to reach a certain value.

4. Spread of Diseases: Exponential functions can model the spread of infectious diseases. The number of infected individuals can grow exponentially in the early stages of an outbreak. Solving exponential equations helps predict the trajectory of the disease and inform public health measures.

5. Cooling and Heating: Newton's Law of Cooling states that the rate of change of an object's temperature is proportional to the difference between its own temperature and the ambient temperature. This law is modeled by an exponential equation, which can be used to determine how long it takes for an object to cool or heat to a certain temperature.

Conclusion

Solving exponential equations is a fundamental skill in mathematics with broad applications across various scientific and practical domains. By mastering the techniques outlined in this guide, including isolating the exponential term, applying logarithms, and utilizing logarithmic properties, you can confidently tackle a wide array of exponential problems. Remember to pay close attention to the details, avoid common pitfalls, and practice regularly to reinforce your understanding. The solution to the equation ${ e^x - 3 = 2 }$, rounded to the nearest thousandth, is approximately 1.609. This example serves as a foundation for solving more complex exponential equations and appreciating their significance in real-world applications. Through continued practice and a solid grasp of the underlying principles, you can excel in solving exponential equations and harnessing their power in various contexts.