Problem 73 - College Algebra with Corequisite Support

Question

For the following exercises, use a calculator to solve the equation. Unless indicated otherwise, round all answers to the nearest ten-thousandth. \(7 e^{3 x-5}+7.9=47\)

Step-by-Step Solution

Answer

\( x \approx 2.2405 \)

Step 1: Isolate the Exponential Term

Start by isolating the exponential term on one side of the equation. Subtract 7.9 from both sides to get the following equation: \[ 7 e^{3x - 5} = 47 - 7.9 \] This simplifies to: \[ 7 e^{3x - 5} = 39.1 \]

Step 2: Divide to Solve for the Exponential

Divide both sides by 7 to isolate the exponential expression: \[ e^{3x - 5} = \frac{39.1}{7} \] Calculate this division to find: \[ e^{3x - 5} \approx 5.5857 \]

Step 3: Apply the Natural Logarithm

Take the natural logarithm (ln) of both sides to solve for the expression in the exponent: \[ \ln(e^{3x - 5}) = \ln(5.5857) \] Using the property \( \ln(e^y) = y \), this simplifies to: \[ 3x - 5 = \ln(5.5857) \]

Step 4: Calculate the Natural Logarithm

Calculate \( \ln(5.5857) \) using a calculator: \[ \ln(5.5857) \approx 1.7215 \] Thus, the equation becomes: \[ 3x - 5 = 1.7215 \]

Step 5: Solve for x

Solve for \( x \) by first adding 5 to both sides: \[ 3x = 1.7215 + 5 \] Simplify to find: \[ 3x = 6.7215 \] Next, divide both sides by 3: \[ x = \frac{6.7215}{3} \approx 2.2405 \]

Key Concepts

Natural Logarithm

The natural logarithm, denoted by \( \ln \), is a special type of logarithm with a base of \( e \), where \( e \) is approximately 2.71828. It's used to solve equations involving exponential functions when the exponent must be isolated. The main property of the natural logarithm used in solving exponential equations like the one in our example is that \( \ln(e^y) = y \). This tells us that taking the natural logarithm of an exponent cancels the base \( e \), leaving us with the exponent itself.

The process of applying \( \ln \) to both sides of the equation transforms an exponential equation into a linear one, making it easier to solve for the desired variable. Using a calculator to find \( \ln(5.5857) \) in our exercise allowed us to determine the value of the unknown exponent.

Isolating Exponential Terms

When solving exponential equations, the first step is to isolate the exponential term. This involves rearranging the equation so that the term with the exponent is on one side, free from any other terms. In the given exercise, we started by subtracting 7.9 from both sides of the equation to get \[ 7 e^{3x - 5} = 39.1 \].

The next step was to divide by the coefficient (7 in this case) of the exponential expression to fully isolate \( e^{3x - 5} \). The result was \[ e^{3x - 5} \approx 5.5857 \]. This clear isolation simplifies the equation, making the subsequent steps of solving much more straightforward. Isolating the exponential term allows us to effectively use the natural logarithm in the next steps.

Calculator Use in Algebra

Calculators are invaluable tools for handling complex calculations in algebra, especially when dealing with non-integer values or irrational numbers like \( e \) and logarithms. In our example, the calculator was used to perform several key computations:

• Dividing 39.1 by 7 to find the value of the exponential expression, giving us \( 5.5857 \)
• Calculating the natural logarithm \( \ln(5.5857) \), providing the approximate result 1.7215
• Finally, dividing 6.7215 by 3 to find \( x \approx 2.2405 \)

Using a calculator not only ensures accuracy but also speeds up the process of solving such equations. It helps to focus on understanding the steps and maintaining precision with more complex values.

Rounding in Calculations

Rounding is a critical part of many algebraic solutions to provide a user-friendly answer that is precise yet manageable. In calculations involving irrational numbers or complex divisions, the results are often long and difficult to use. For the exercise, it was necessary to round answers to four decimal places, especially when dealing with logarithms and divisions.

For instance, after calculating the division \( \frac{39.1}{7} \) and finding \( e^{3x - 5} \approx 5.5857 \), the number was rounded to enhance readability and usability. Additionally, \( \ln(5.5857) \approx 1.7215 \) was rounded to four decimal places as well as the final value of \( x \approx 2.2405 \).

Rounding should be done carefully to ensure mathematical accuracy is maintained, even when simplifying complex results with long decimals.