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Chapter 4: Problem 30

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. $$9 e^{x}=107$$

Short Answer

Expert verified
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 by step solution

01

Isolate the exponential expression

Divide both sides of the equation by 9 to isolate \( e^{x} \). The equation becomes \( e^{x} = 107/9 \).
02

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 \( x = \ln(107/9) \).
03

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 \( x \).

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