Solve for x 3+2e^(-x)=6

Problem

3+2*e(−x)=6

Solution

1. Isolate the exponential term by subtracting 3 from both sides of the equation.

2*e(−x)=3

2. Divide both sides by 2 to further isolate the exponential expression.

e(−x)=3/2

3. Apply the natural logarithm to both sides to eliminate the base e

ln(e(−x))=ln(3/2)

4. Simplify the left side using the property ln(eu)=u

−x=ln(3/2)

5. Solve for x by multiplying both sides by −1

x=−ln(3/2)

6. Rewrite the expression using logarithmic properties if desired, noting that −ln(a)=ln(a(−1))

x=ln(2/3)

Final Answer

x=ln(2/3)

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