Problem 29 - College Algebra Essentials

Question

Solve each exponential equation. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$5 e^{x}=23$$

Step-by-Step Solution

Answer

The solution to the equation is \(x \approx 1.33\). This is obtained by evaluating \(\ln(\frac{23}{5})\) and rounding the result to two decimal places.

Step 1: Isolate the Exponential Function

The first step is to isolate \(e^{x}\) on one side of the equation. To do this, divide both sides of the equation by 5. This gives us the equation in the form \(e^{x} = \frac{23}{5}\).

Step 2: Apply Natural Logarithm Function to Both Sides

Next, apply the natural logarithm function to both sides of the equation. Note that the natural logarithm is the inverse function of the exponential function. This will give us the equation: \(\ln(e^{x}) = \ln(\frac{23}{5})\). This simplifies to \(x = \ln(\frac{23}{5})\).

Step 3: Calculate Numerical Approximation

Finally, to obtain a decimal approximation for the solution, use a calculator to find the value of \(\ln(\frac{23}{5})\). Ensure the value is correct to two decimal places.