Solving 5e^(2x) - 4 = 11 A Step By Step Explanation

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In the realm of mathematics, exponential equations hold a significant place, often appearing in various scientific and engineering applications. These equations involve exponential functions, where the variable appears in the exponent. Solving exponential equations requires a systematic approach, employing the properties of logarithms and exponential functions to isolate the variable. In this comprehensive guide, we will delve into the process of solving the exponential equation 5e^(2x) - 4 = 11, providing a step-by-step explanation to ensure a clear understanding of the solution. Furthermore, we will explore the underlying concepts and techniques involved in solving exponential equations, empowering you to tackle similar problems with confidence.

Understanding Exponential Equations

Exponential equations are mathematical expressions where the variable appears in the exponent of a term. These equations often model phenomena that exhibit exponential growth or decay, such as population dynamics, radioactive decay, and compound interest. The general form of an exponential equation is:

a^(f(x)) = b

where:

  • a is the base, a positive real number not equal to 1
  • f(x) is a function of the variable x
  • b is a constant

Solving exponential equations involves isolating the variable x. This typically requires using the properties of logarithms to undo the exponential function. Logarithms are the inverse functions of exponentials, and they allow us to bring the exponent down as a coefficient.

Step 1: Isolate the Exponential Term

The initial step in solving the equation 5e^(2x) - 4 = 11 is to isolate the exponential term, e^(2x). This involves performing algebraic manipulations to get the exponential term by itself on one side of the equation.

To isolate e^(2x), we first add 4 to both sides of the equation:

5e^(2x) - 4 + 4 = 11 + 4

This simplifies to:

5e^(2x) = 15

Next, we divide both sides of the equation by 5:

(5e^(2x))/5 = 15/5

This gives us:

e^(2x) = 3

Now, the exponential term e^(2x) is isolated on one side of the equation.

Step 2: Apply the Natural Logarithm

To solve for x, we need to undo the exponential function. This is where logarithms come into play. Since the base of the exponential term is e, the natural logarithm (ln) is the appropriate logarithm to use. The natural logarithm is the logarithm to the base e.

We take the natural logarithm of both sides of the equation:

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

Using the property of logarithms that ln(a^b) = b * ln(a), we can simplify the left side of the equation:

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

Since ln(e) = 1, the equation becomes:

2x = ln(3)

Step 3: Solve for x

Finally, to solve for x, we divide both sides of the equation by 2:

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

This gives us the solution:

x = ln(3)/2

Therefore, the solution to the exponential equation 5e^(2x) - 4 = 11 is x = ln(3)/2, which corresponds to option C.

General Strategies for Solving Exponential Equations

The approach we used to solve the equation 5e^(2x) - 4 = 11 can be generalized to solve a wide range of exponential equations. Here are some key strategies to keep in mind:

  1. Isolate the Exponential Term: The first step is always to isolate the exponential term on one side of the equation. This may involve adding, subtracting, multiplying, or dividing both sides of the equation by appropriate constants.
  2. Apply Logarithms: Once the exponential term is isolated, take the logarithm of both sides of the equation. The choice of logarithm depends on the base of the exponential term. If the base is e, use the natural logarithm (ln). If the base is 10, use the common logarithm (log). In general, if the base is b, use logarithm to the base b.
  3. Use Logarithm Properties: Apply the properties of logarithms to simplify the equation. The most commonly used properties include:
    • ln(a^b) = b * ln(a)
    • ln(a * b) = ln(a) + ln(b)
    • ln(a / b) = ln(a) - ln(b)
  4. Solve for the Variable: After applying logarithm properties, solve the resulting equation for the variable. This may involve algebraic manipulations such as adding, subtracting, multiplying, or dividing both sides of the equation.

Example 2: Solving a More Complex Exponential Equation

Let's consider a slightly more complex exponential equation:

3^(2x + 1) = 27

  1. Isolate the Exponential Term: The exponential term is already isolated on the left side of the equation.

  2. Apply Logarithms: We can take the logarithm of both sides of the equation. Since 27 is a power of 3, it is easiest to use the logarithm with base 3 in this case. However, we can also use the natural logarithm (ln) or the common logarithm (log).

    Using the natural logarithm, we get:

    ln(3^(2x + 1)) = ln(27)

  3. Use Logarithm Properties: Apply the property ln(a^b) = b * ln(a):

    (2x + 1) * ln(3) = ln(27)

  4. Solve for the Variable: Divide both sides by ln(3):

    (2x + 1) = ln(27) / ln(3)

    Since 27 = 3^3, ln(27) = ln(3^3) = 3 * ln(3). Therefore:

    (2x + 1) = (3 * ln(3)) / ln(3) = 3

    Subtract 1 from both sides:

    2x = 2

    Divide by 2:

    x = 1

Therefore, the solution to the exponential equation 3^(2x + 1) = 27 is x = 1.

Conclusion

Solving exponential equations is a fundamental skill in mathematics with applications across various fields. By understanding the properties of logarithms and exponential functions, and by following a systematic approach, you can confidently solve a wide range of exponential equations. Remember to isolate the exponential term, apply logarithms, use logarithm properties, and solve for the variable. With practice, you will master the art of solving exponential equations and unlock their power in mathematical problem-solving.

In summary, the solution to the equation 5e^(2x) - 4 = 11 is x = ln(3)/2, which we found by isolating the exponential term, applying the natural logarithm, and solving for x. This process highlights the key steps in solving exponential equations and provides a foundation for tackling more complex problems in the future.