Solve for x e^(ax)=Ce^(bx)

Problem

e(a*x)=C*e(b*x)

Solution

1. Divide both sides of the equation by e(b*x) to group the exponential terms on one side.

(e(a*x))/(e(b*x))=C

2. Apply the quotient rule for exponents, which states that (eu)/(ev)=e(u−v)

e(a*x−b*x)=C

3. Factor out the common variable x from the exponent.

e(x*(a−b))=C

4. Take the natural logarithm of both sides to eliminate the exponential base e

ln(e(x*(a−b)))=ln(C)

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

x*(a−b)=ln(C)

6. Isolate x by dividing both sides by the term (a−b)

x=ln(C)/(a−b)

Final Answer

x=ln(C)/(a−b)

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