Solving E^(4x) = 16 A Step-by-Step Guide
Solve the equation e^(4x) = 16 for x.
Introduction
In this comprehensive guide, we will delve into the process of solving for the variable x in the equation e^(4x) = 16. This type of equation falls under the category of exponential equations, where the variable appears in the exponent. Understanding how to solve such equations is fundamental in various fields, including mathematics, physics, engineering, and finance. We will explore the underlying concepts, step-by-step methods, and provide clear explanations to ensure a solid understanding of the solution. Our main keywords are solving exponential equations, finding x, e^(4x) = 16. This detailed exploration will not only provide the solution to the given problem but also equip you with the skills to tackle similar exponential equations effectively.
Understanding Exponential Equations
Before diving into the specifics of solving e^(4x) = 16, it's crucial to understand the nature of exponential equations. An exponential equation is an equation in which the variable occurs in the exponent. The general form of an exponential equation is a^(f(x)) = b, where a and b are constants and f(x) is a function of x. The key to solving exponential equations lies in the properties of exponents and logarithms. Specifically, logarithms provide a way to “undo” exponentiation, allowing us to isolate the variable in the exponent. The natural exponential function, denoted by e^x, is a particularly important case in calculus and many applications. The number e is an irrational number approximately equal to 2.71828, and the natural logarithm, denoted by ln(x), is the logarithm to the base e. The natural logarithm is the inverse function of the natural exponential function, meaning that ln(e^x) = x and e^(ln(x)) = x. These properties are essential for solving equations involving e. Recognizing these fundamental principles is the first step in effectively solving for x in equations like e^(4x) = 16. Mastering exponential equations opens doors to solving real-world problems related to growth, decay, and various natural phenomena. Therefore, a thorough understanding of the underlying concepts is paramount.
Step-by-Step Solution
To solve the equation e^(4x) = 16, we will employ the following steps:
Step 1: Take the Natural Logarithm of Both Sides
The crucial step in solving exponential equations involving e is to take the natural logarithm (ln) of both sides of the equation. This is because the natural logarithm is the inverse function of the exponential function with base e. Applying the natural logarithm to both sides of the equation e^(4x) = 16, we get:
ln(e^(4x)) = ln(16)
This step is vital because it allows us to utilize the property of logarithms that states ln(a^b) = b * ln(a). In our case, this property will help us to bring the exponent 4x down as a coefficient, making it easier to isolate x. The natural logarithm effectively “undoes” the exponential function, simplifying the equation significantly. Understanding this fundamental step is key to solving a wide range of exponential equations. The ability to correctly apply the natural logarithm is a cornerstone in dealing with exponential functions and their applications.
Step 2: Apply the Logarithmic Property ln(e^u) = u
Using the property that the natural logarithm of e raised to any power u is equal to u, i.e., ln(e^u) = u, we can simplify the left side of the equation. Applying this property to our equation ln(e^(4x)) = ln(16), we get:
4x = ln(16)
This simplification is a direct application of the inverse relationship between the natural exponential function and the natural logarithm. The natural logarithm effectively “cancels out” the exponential function with base e, leaving us with just the exponent. This step is critical because it transforms the exponential equation into a simple algebraic equation that is much easier to solve. The ability to recognize and apply this property is essential for efficiently solving exponential equations. Understanding the inverse relationship between exponential and logarithmic functions is a fundamental concept in mathematics.
Step 3: Solve for x
Now that we have the equation 4x = ln(16), we can easily solve for x by dividing both sides of the equation by 4:
x = ln(16) / 4
This step isolates x, giving us an expression for its value in terms of the natural logarithm of 16. However, it's often helpful to simplify ln(16) further before calculating the final numerical value of x. Recognizing opportunities for simplification is an important skill in mathematics, as it can often lead to a more elegant and understandable solution. In this case, we can simplify ln(16) by expressing 16 as a power of 2, which will allow us to use additional logarithmic properties. This approach not only makes the calculation easier but also provides a deeper understanding of the relationships between numbers and their logarithms. Therefore, this step is a crucial part of the problem-solving process.
Step 4: Simplify ln(16)
We can simplify ln(16) by expressing 16 as a power of 2. Since 16 = 2^4, we can rewrite ln(16) as ln(2^4). Using the logarithmic property ln(a^b) = b * ln(a), we have:
ln(16) = ln(2^4) = 4 * ln(2)
Substituting this back into our equation for x, we get:
x = (4 * ln(2)) / 4
This simplification is a key step in finding the exact value of x. By expressing 16 as a power of 2, we are able to utilize the properties of logarithms to simplify the expression. This technique is widely applicable in solving logarithmic and exponential equations. Recognizing and applying these properties is a crucial skill in mathematics. The ability to manipulate logarithmic expressions effectively often leads to simpler and more manageable equations. This step not only simplifies the calculation but also provides a deeper understanding of logarithmic relationships.
Step 5: Final Solution
Now, we can simplify the expression for x by canceling the common factor of 4:
x = (4 * ln(2)) / 4 = ln(2)
Thus, the solution for x is ln(2). This is the exact value of x. If you need a decimal approximation, you can use a calculator to find that ln(2) ≈ 0.693. Therefore, the solution is approximately x ≈ 0.693. Arriving at the final solution involves careful simplification and application of logarithmic properties. This process underscores the importance of understanding the relationships between exponential and logarithmic functions. The exact solution, x = ln(2), provides a precise representation of the value of x, while the decimal approximation offers a practical understanding of its magnitude. This comprehensive approach ensures a thorough understanding of the solution and its significance. The ability to solve for x in such equations is fundamental in many areas of mathematics and its applications.
Alternative Methods
While using the natural logarithm is the most straightforward method for solving e^(4x) = 16, there are alternative approaches that can provide additional insights and reinforce understanding. One such method involves using the general logarithm with a different base. Let's explore this alternative method.
Using the General Logarithm
Instead of taking the natural logarithm (base e), we could take the logarithm with any base. For example, we can use the base 10 logarithm, denoted as log10 or simply log. Applying the base 10 logarithm to both sides of the equation e^(4x) = 16, we get:
log(e^(4x)) = log(16)
Now, using the logarithmic property log(a^b) = b * log(a), we have:
4x * log(e) = log(16)
Solving for x, we get:
x = log(16) / (4 * log(e))
This expression for x is equivalent to our previous solution, x = ln(16) / 4, but it is expressed in terms of base 10 logarithms. To see this equivalence, we can use the change of base formula, which states that log_b(a) = ln(a) / ln(b). Applying this formula to the base 10 logarithm, we have:
log(e) = ln(e) / ln(10) = 1 / ln(10)
log(16) = ln(16) / ln(10)
Substituting these expressions back into our equation for x, we get:
x = (ln(16) / ln(10)) / (4 * (1 / ln(10)))
x = ln(16) / (4 * ln(10)) * ln(10)
x = ln(16) / 4
This is the same expression we obtained using the natural logarithm method. Thus, using the general logarithm method confirms our previous solution. This alternative approach demonstrates the flexibility of logarithmic properties and provides a deeper understanding of the relationship between logarithms of different bases. It also reinforces the fundamental principle that the choice of base does not affect the final solution, as long as the logarithmic properties are applied correctly.
Conclusion
In summary, we have thoroughly explored how to solve the equation e^(4x) = 16. We began by understanding the nature of exponential equations and the key properties of logarithms. We then systematically applied the natural logarithm to both sides of the equation, utilized the property ln(e^u) = u, and solved for x. We also simplified ln(16) using logarithmic properties to arrive at the exact solution, x = ln(2). Furthermore, we discussed an alternative method using the general logarithm, which reinforced our understanding of logarithmic principles and demonstrated the equivalence of different approaches. The solution x = ln(2) is a fundamental result and underscores the importance of mastering exponential and logarithmic equations. These types of equations appear frequently in various mathematical and scientific contexts, making the ability to solve them a crucial skill. By understanding the underlying concepts and practicing the step-by-step methods, you can confidently tackle similar problems and expand your mathematical proficiency. The key takeaway is the power of logarithms in “undoing” exponentiation and simplifying complex equations.