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Chapter 3: Problem 55

Solve the exponential equation algebraically. Approximate the result to three decimal places. $$7-2 e^{x}=5$$

Short Answer

Expert verified
The solution to the equation is \(x = 0\).

Step by step solution

01

Rewrite the equation

Subtract 7 from both sides to isolate the term with the exponential: \(7 - 2 e^{x} - 7 = 5 - 7\). This gives: \(-2 e^{x} = -2\).
02

Isolate the exponential term

Divide both sides by -2: \(\(-2 e^{x} / -2 = -2 / -2\)\). This simplifies to \(e^{x} = 1\)
03

Apply the natural logarithm

Apply the natural logarithm (ln) to both sides to get rid of the exponential: \(\ln(e^{x}) = \ln(1)\). Simplifying this gives \(x = \ln(1)\)
04

Solve for x

Recall that \(\ln(1) = 0\), so the solution to the equation is \(x = 0\)
05

Approximate the result

Since the result is exactly 0, there's no need to approximate it to three decimal places. The solution remains as \(x = 0\)

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