Problem 527 - Precalculus

Question

Find the exact solution for \(5 e^{3 x}-4=6 .\) If there is no solution, write no solution.

Step-by-Step Solution

Answer

The exact solution is \(x = \frac{\ln(2)}{3}\).

Step 1: Isolate the Exponential Term

Start with the equation: \[ 5e^{3x} - 4 = 6 \] Add 4 to both sides to isolate the exponential expression:\[ 5e^{3x} = 10 \]

Step 2: Solve for the Exponent

Divide both sides by 5 to further isolate the exponential term:\[ e^{3x} = 2 \]

Step 3: Apply the Natural Logarithm

Take the natural logarithm (ln) of both sides to solve for the exponent:\[ \ln(e^{3x}) = \ln(2) \]Since \( \ln(e^{y}) = y \); therefore,\[ 3x = \ln(2) \]

Step 4: Solve for x

Divide both sides by 3 to solve for \( x \):\[ x = \frac{\ln(2)}{3} \]

Step 5: Final Calculation

Use a calculator to find the value of \( \ln(2) \) if necessary for further evaluation, or leave as it is for an exact solution. The solution is:\[ x \approx 0.231 \] using the approximation for \( \ln(2) \) calculated as approximately 0.693.

Key Concepts

Natural Logarithm

The natural logarithm, often denoted as ln, is a mathematical function that is used to transform exponential functions into linear ones, making them easier to solve. It is the inverse operation of the exponential function with base e, where e is approximately 2.718.
Understanding the natural logarithm is crucial when dealing with exponential equations since it allows us to isolate and solve for the exponent in the equation. For example, in the equation \( e^{3x} = 2 \), applying the natural logarithm to both sides gives \( \ln(e^{3x}) = \ln(2) \). This step is particularly important because it transforms the power of e, the base, into a more manageable linear form, \( 3x = \ln(2) \).
Some key properties to remember about the natural logarithm include:

• \( \ln(e^x) = x \)
• \( \ln(1) = 0 \)
• \( \ln(ab) = \ln(a) + \ln(b) \)

By mastering these properties, you can more easily manipulate equations and understand the role of the natural logarithm in various contexts.

Solving Equations

Solving equations involves finding the value of unknown variables that make the equation true. When dealing with exponential equations, like \(5 e^{3 x} - 4 = 6\), it’s essential to understand how to systematically isolate the variable to unveil the solution.
The first step typically involves isolating the term with the exponent. For instance, adding 4 to both sides of the initial equation \(5 e^{3x} - 4 = 6\) results in \(5 e^{3x} = 10\), which simplifies the problem and gets it closer to a solvable form.
Next, divide through by the coefficient of the exponential expression. In this example, dividing both sides by 5 gives \(e^{3x} = 2\). The equation is now in a form where taking the natural logarithm is the logical next step.

• Isolate the exponential or variable term
• Use inverse operations like division or square root where applicable
• Finally, apply logarithms to handle exponential terms effectively

Throughout the process, maintaining balance by performing the same operation on both sides of the equation is fundamental to solving equations accurately.

Exact Solutions

An exact solution refers to the precise mathematical answer to a given equation without approximations. In exercises involving exponential equations, obtaining an exact solution often implies leaving the answer in logarithmic form rather than a decimal or rounded approximation.
For the equation we solved, \( x = \frac{\ln(2)}{3} \) is the exact solution. This form represents the exact numerical value of x without converting the natural logarithm to its decimal equivalent, allowing it to remain in its pure mathematical state.
Exact solutions are particularly valuable in theoretical mathematics and when further symbolic calculations are needed, as they offer the utmost precision. When decimal values are necessary, approximation can be used as in \( x \approx 0.231 \), but this represents an approximate solution derived from \( \ln(2) \approx 0.693 \).

• Exact solutions preserve purity and precision
• Utilize symbolic representation wherever possible
• Approximate only when necessary

Understanding the importance of exact solutions enhances your problem-solving ability and enables you to appreciate the depth and accuracy required in mathematical analysis.