Solve for x 1000e^(-4x)=75

Problem

1000*e(−4*x)=75

Solution

1. Isolate the exponential term by dividing both sides of the equation by 1000

e(−4*x)=75/1000

2. Simplify the fraction on the right side by dividing the numerator and denominator by their greatest common divisor, 25

e(−4*x)=3/40

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

ln(e(−4*x))=ln(3/40)

4. Use the inverse property of logarithms, ln(eu)=u to simplify the left side.

−4*x=ln(3/40)

5. Solve for x by dividing both sides by −4

x=ln(3/40)/(−4)

6. Rewrite the expression using logarithm properties, specifically ln(a/b)=ln(a)−ln(b) or by moving the negative sign.

x=(ln(40)−ln(3))/4

Final Answer

x=ln(3/40)/(−4)

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