Question

Solve the exponential equation algebraically. Approximate the result to three decimal places.\(500 e^{-x}=300\)

Short answer

The solution of the equation is \(x = 0.511\)

Step-by-step solution

Isolate the exponential expression

Divide both sides of the equation by 500 to isolate the exponential expression. This will leave us with \(e^{-x}=\frac{300}{500}\) which simplifies to \(e^{-x}=0.6\)

Apply natural logarithm

Apply natural logarithm on both sides to remove the exponential part. It will result in \(-x = ln(0.6)\)

Solve for the variable \(x\)

By multiplying both sides with -1, we can solve for the variable \(x\). This will give \(x = -ln(0.6)\)

Approximate the result

Plug ln(0.6) into the calculator and multiply by -1 to get the value of \(x\). The value should be approximated to three decimal places.

Key concepts

Natural Logarithm

In solving exponential equations, especially those involving the mathematical constant 'e' (approximately 2.718), the natural logarithm (ln) plays a crucial role. The natural logarithm is the inverse operation of the exponential function with base 'e'. This means that if you have an equation such as \( e^y = x \), taking the natural logarithm of both sides gives you \( y = \ ln(x) \). This operation essentially helps you "unravel" the exponential, allowing you to solve for the exponent or variable.

In our exercise, we have \( e^{-x} = 0.6 \). To eliminate the 'e' and isolate the exponent, we apply a natural logarithm to both sides of the equation. This transforms it into \( -x = \ln(0.6) \). The ln function is especially useful here because it seamlessly handles calculations involving 'e', simplifying complex expressions during problem-solving.

Isolate the Variable

When faced with equations, especially exponential ones, isolating the variable is a crucial step in finding its value. The goal is to get the variable by itself on one side of the equation, so you can then solve it more straightforwardly.

In the given exercise, we started with the equation \( 500e^{-x} = 300 \). Initially, we have both numeric values and the exponential term on one side. To isolate the exponential expression \( e^{-x} \), we divide both sides by 500. This simple division transforms the equation to \( e^{-x} = 0.6 \), effectively simplifying the task, and setting up the next step.

By reducing the equation to this form, you ensure that the variable you need to solve for becomes more apparent, making it easier to handle in subsequent logarithmic operations.

Approximation Techniques

Often, solutions to equations are not whole numbers or easy-to-calculate values. In such cases, approximation becomes essential, especially when you need a practical answer with a certain degree of precision.

For this exercise, after isolating the variable and applying the natural logarithm, we obtained \( x = - \ln(0.6) \). While this is correct, it's essential to provide a numerical approximation, typically with a calculator, to make it more usable in real-world applications.

Key steps in approximation:

• Use a calculator to compute \( \ln(0.6) \).
• Multiply the result by -1 to account for the negative sign associated with \(-x\).
• Round or adjust the answer to three decimal places to meet the requirement of accuracy demanded by the problem.

By following these steps, you ensure that your final result is both precise and presented in a format that's readily applicable, providing a clearer understanding and confidence in your solution.