Problem 25 - 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. $$e^{x}=5.7$$

Step-by-Step Solution

Answer

The solution for the equation \(e^{x}=5.7\) is \(x = \ln{(5.7)} \approx 1.74\).

Step 1: Take natural log on both sides

The power e can be eliminated by taking natural logarithm at both sides. Hence,\(\ln(e^{x}) = \ln{(5.7)}\).

Step 2: Apply the property of logarithm

The property of logarithm states that when a logarithm has an exponent, you can rewrite this exponent as a factor at the front. So, \(x\ln{(e)} = \ln{(5.7)}\). Here, the natural logarithm of \(e\) is \(1\), so you get the equation \(x= \ln{(5.7)}\)

Step 3: Calculate the decimal approximation

Using a calculator, calculate the decimal approximation of \(\ln{(5.7)}\). It should be \(1.74\) to two decimal places.