Problem 36 - 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. $$e^{4 x-5}-7=11,243$$

Step-by-Step Solution

Answer

The solution to the exponential equation \(e^{4x-5} = 11,243\) is \(x \approx 3.20\).

Step 1: Simplify the equation

First, add 7 to both sides of the equation to isolate the exponential term. This gives: \(e^{4x-5} = 11,250\).

Step 2: Apply natural logarithm

In order to isolate \(x\), take the natural logarithm (ln) on both sides. This yields: \(ln(e^{4x-5}) = ln(11250)\). The left side simplifies using the property of logarithms \(ln(e^{a}) = a\), resulting in: \(4x - 5 = ln(11250)\).

Step 3: Solve for x

Next, solve for \(x\) by first adding 5 to both sides and then dividing by 4. This leads to the solution: \(x = (ln(11250) + 5) / 4\).

Step 4: Find decimal approximation

Finally, using a calculator, approximate this solution to two decimal places: \(x \approx 3.20\).