Problem 15 - Algebra for College Students

Question

Solve each exponential equation and express approximate solutions to the nearest hundredth. $$ 3 e^{x}-1=17 $$

Step-by-Step Solution

Answer

\( x \approx 1.79 \)

Step 1: Isolate the Exponential Term

Start with the equation \( 3e^x - 1 = 17 \).Add 1 to both sides to isolate the exponential expression: \( 3e^x = 18 \).

Step 2: Divide to Solve for Exponential Term

Divide both sides by 3 to completely isolate the term with the exponent:\( e^x = 6 \).

Step 3: Take the Natural Logarithm

Apply the natural logarithm to both sides to solve for \( x \):\( \ln(e^x) = \ln(6) \).This simplifies to \( x = \ln(6) \) because the natural logarithm and the exponential function are inverse operations.

Step 4: Compute the Value

Use a calculator to find an approximate value of \( x \):\( x \approx \ln(6) \approx 1.79 \), rounded to the nearest hundredth.

Key Concepts

Natural Logarithm

The natural logarithm, commonly denoted as \( \ln(x) \), is a logarithm with the base \( e \), where \( e \) is approximately equal to 2.718. This special logarithm is very useful in calculus and exponential equations because it simplifies working with exponential functions.

• A defining feature of \( \ln \) is that \( \ln(e) = 1 \) because the power of \( e \) to get \( e \) itself is 1.
• The natural logarithm and the exponential function are inverse operations. This means that \( \ln(e^x) = x \) and \( e^{\ln(x)} = x \).

In our example, we took the natural logarithm of both sides of the equation \( e^x = 6 \) in order to solve for \( x \). This step involved using the property that \( \ln(e^x) = x \), which allowed us to simplify the left side of the equation to just \( x \). Thus, solving the equation \( \ln(e^x) = \ln(6) \) becomes much more straightforward.

Isolating Exponential Terms

Isolating the exponential term is an essential step when solving exponential equations. This step involves rearranging the equation so that the exponential expression stands alone on one side.

• First, we removed any constant terms from the side with the exponential. In our example, we started with \( 3e^x - 1 = 17 \). By adding 1 to both sides, we simplified this to \( 3e^x = 18 \).
• Next, to further isolate the exponential term \( e^x \), divide both sides by the coefficient of the exponential term. For our example: dividing by 3 gives us \( e^x = 6 \).

These actions are crucial because they set the stage for applying logarithmic operations. Once isolated, it's easier to proceed with solving the equation.

Approximating Solutions

After isolating the exponential term and applying the natural logarithm, the next step is to find an approximate solution. Often, exact solutions involve numbers like \( \ln(6) \) that need further calculation to convert them to a more manageable form.

• Using a calculator or computational tool, you can find that \( \ln(6) \approx 1.79 \). This is because calculators are designed to handle complex mathematical functions, returning results with high precision.
• The question asked for solutions to be rounded to the nearest hundredth. The precise outcome from \( \ln(6) \) when computed results in \( x \approx 1.79 \), fulfilling this requirement.

Rounding allows for easier interpretation and practical use of solutions, especially in contexts where excessive precision might not be necessary. It's always important to follow any instruction in a problem about how to round or approximate results.