Problem 16 - Algebra and Trigonometry

Question

Find the solution of the exponential equation, correct to four decimal places. $$ e^{3-5 x}=16 $$

Step-by-Step Solution

Answer

The solution is \(x = 0.0455\).

Step 1: Take the natural logarithm of both sides

The equation given is \(e^{3-5x}=16\). To eliminate the exponential function on the left side, we take the natural logarithm (\(\ln\)) of both sides of the equation, which gives us: \(\ln(e^{3-5x}) = \ln(16)\).

Step 2: Use the property of logarithms

Using the property of logarithms that states \(\ln(e^y) = y\), we can simplify the left side of the equation to \(3 - 5x = \ln(16)\).

Step 3: Solve for \(x\)

Now we need to solve the equation \(3 - 5x = \ln(16)\) for \(x\). First, compute \(\ln(16)\) which is approximately 2.7726. So, the equation becomes \(3 - 5x = 2.7726\).

Step 4: Rearrange the equation for \(x\)

Subtract 3 from both sides to get \(-5x = 2.7726 - 3\), which simplifies to \(-5x = -0.2274\).

Step 5: Solve for \(x\)

Divide both sides by -5 to isolate \(x\):\[x = \frac{-0.2274}{-5}\] which simplifies to \(x = 0.0455\).

Key Concepts

Natural Logarithms

Natural logarithms are a type of logarithm that use the base of Euler's number, denoted as \( e \). The constant \( e \) is approximately equal to 2.71828, and it arises naturally in various mathematical contexts, especially in continuous growth processes. The natural logarithm of a number \( x \) is written as \( \ln(x) \). It is the inverse function of the exponential function with the base \( e \).

When you apply the natural logarithm to an exponential equation like \( e^{3-5x} = 16 \), you're essentially performing a transformation that helps "undo" the exponential, making the expression easier to work with. Thus, \( \ln(e^{3-5x}) = 3 - 5x \). This property allows us to extract the exponent, which is crucial for solving equations where the variable is in an exponent.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. For exponential equations, such as \( e^{3-5x} = 16 \), we often use logarithms to isolate the variable.

The strategy involves several steps:

• First, take the natural logarithm of both sides of the equation, which simplifies the exponential expression.
• Next, use logarithmic properties to simplify the expression further.
• Once the equation is in a simpler form, solve algebraically for the unknown variable by isolating it on one side of the equation.

In our exercise, after applying logarithms, the equation becomes \(3 - 5x = \ln(16)\). From here, it becomes a matter of simple algebra to solve for \(x\).

Logarithmic Properties

Logarithmic properties are crucial helpers when working with equations involving logarithms and exponentials. They allow for significant simplifications, making complex expressions more manageable.

Three key properties often used include:

• \(\ln(e^y) = y\): This property helped us simplify \( \ln(e^{3-5x}) \) to \( 3 - 5x \) in the step-by-step process.
• Product Property: \(\ln(a \cdot b) = \ln(a) + \ln(b)\).
• Quotient Property: \(\ln(\frac{a}{b}) = \ln(a) - \ln(b)\).

Each of these properties can transform an otherwise tricky equation into something solvable. In this case, the simplicity derived from the first property was key to solving the exercise.

Decimal Approximation

When dealing with logarithms and equations, decimal approximation is often used to obtain a practical and precise numerical answer. Exact solutions can be cumbersome, so we simplify by calculating to a specific decimal place.

In the problem, we approximated \( \ln(16) \) to 2.7726, which is precise to four decimal places. Decimal approximation is particularly important because:

• It allows us to handle irrational numbers easily.
• It ensures clarity in communicating numerical solutions.
• It gives a serviceable precision without compromising on significant details.

Thus, when `x` becomes approximately 0.0455, this practice ensures the solution is clear and usable, especially in applied contexts where exactness might be secondary to practicality.