Problem 43 - Algebra 2

Question

Solve each equation. Round to the nearest ten-thousandth. $$ 3 e^{x}-2=0 $$

Step-by-Step Solution

Answer

\( x \approx -0.4055 \)

Step 1: Isolate the Exponential Expression

Start by isolating the term with the exponential expression. In the equation \( 3e^{x} - 2 = 0 \), add 2 to both sides to get: \[ 3e^{x} = 2 \].

Step 2: Solve for the Exponential Term

Next, divide both sides of the equation by 3 to solve for \( e^{x} \): \[ e^{x} = \frac{2}{3} \].

Step 3: Apply the Natural Logarithm

To eliminate the exponential function, apply the natural logarithm (ln) to both sides of the equation. This gives:\[ \ln(e^{x}) = \ln\left(\frac{2}{3}\right) \]. Using the property \( \ln(e^{x}) = x \), simplify this to:\[ x = \ln\left(\frac{2}{3}\right) \].

Step 4: Compute the Value

Calculate the value of \( \ln\left(\frac{2}{3}\right) \) using a calculator. The result is approximately \( x \approx -0.405465 \).

Step 5: Round to the Nearest Ten-Thousandth

Finally, round the solution to the nearest ten-thousandth. Thus, \( x \approx -0.4055 \).

Key Concepts

Natural Logarithm

A natural logarithm, denoted as \( \ln \), is the logarithm to the base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828. This special type of logarithm is frequently used in solving exponential equations. When we have an equation like \( e^x = a \), we use the natural logarithm to help us "unwrap" \( x \) from the exponential. This is because the natural logarithm and the exponential function are inverse operations. Thus, applying \( \ln \) to both sides converts \( e^x \) into \( x \) using the property \( \ln(e^x) = x \). This is a very helpful property when dealing with equations involving \( e \). In our problem, applying \( \ln \) to \( \frac{2}{3} \) gives \( x = \ln\left(\frac{2}{3}\right) \). This makes it easier to solve for \( x \) after isolating the exponential term.

Rounding Numbers

Rounding numbers is an essential skill in mathematics that helps simplify numbers for interpretation and communication. In this context, rounding to the nearest ten-thousandth means you look four decimal places to the right of the decimal point. If the number in the fifth place (after the fourth decimal) is 5 or greater, the fourth place digit increases by one. If it is less than 5, the fourth place digit remains unchanged. For example, the number -0.405465 needs to be rounded for our solution. We analyze the fifth decimal place, which is 6, indicating that the fourth decimal (5) should increase by 1. Therefore, the final rounded value becomes -0.4055. This process ensures the solution is presented in a more manageable and concise format.

Isolating Variables

Isolating a variable is fundamental in solving equations as it enables you to find the value of the unknown. The process involves manipulating the equation to have the variable on one side and everything else on the other. The key to isolating variables is adhering to the rules of algebra, such as performing the same operation on both sides of the equation. In the given exercise, the goal was to solve for \( x \). Initially, we had the equation \( 3e^x - 2 = 0 \). The steps included:

• Adding 2 to both sides, simplifying the equation to \( 3e^x = 2 \).
• Dividing both sides by 3 to further isolate \( e^x \), giving \( e^x = \frac{2}{3} \).

These steps effectively isolate \( e^x \), allowing further simplification using the natural logarithm to solve for \( x \). Practicing this skill makes complex equations more manageable and builds a solid foundation for solving various math problems.