Problem 30 - 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. $$9 e^{x}=107$$

Step-by-Step Solution

Answer

The solution to the equation \(9 e^{x}=107\) can be expressed in terms of natural logarithms as \(x = \ln(107/9)\). The decimal approximation for this logged value, correct to two decimal places, will then be the value of \(x\).

Step 1: Isolate the exponential expression

Divide both sides of the equation by 9 to isolate \( e^{x} \). The equation becomes \( e^{x} = 107/9 \).

Step 2: Take natural logarithm on both sides

Take the natural logarithm of both sides. The left-hand side becomes \( \ln(e^{x}) \), and the right-hand side becomes \( \ln(107/9) \). Because the natural log of \( e^{x} \) simplifies to \( x \), the equation becomes \( x = \ln(107/9) \).

Step 3: Calculate x value

Use a calculator to calculate the result of \( \ln(107/9) \), rounding to two decimal places. This will give the value of \( x \).