Solve for x 4^(1-x)=3^(2x+5)

Problem

4(1−x)=3(2*x+5)

Solution

1. Take the natural logarithm of both sides of the equation to move the variables out of the exponents.

ln(4(1−x))=ln(3(2*x+5))

2. Apply the power rule for logarithms, which states ln(ab)=b*ln(a) to both sides.

(1−x)*ln(4)=(2*x+5)*ln(3)

3. Distribute the logarithmic constants into the parentheses.

ln(4)−x*ln(4)=2*x*ln(3)+5*ln(3)

4. Group the terms containing x on one side of the equation and the constant terms on the other.

ln(4)−5*ln(3)=2*x*ln(3)+x*ln(4)

5. Factor out the common factor x from the right side.

ln(4)−5*ln(3)=x*(2*ln(3)+ln(4))

6. Solve for x by dividing both sides by the expression in the parentheses.

x=(ln(4)−5*ln(3))/(2*ln(3)+ln(4))

7. Simplify the expression using logarithm properties such as n*ln(a)=ln(an) and ln(a)±ln(b)=ln(a⋅b) or ln(a/b)

x=(ln(4)−ln(3))/(ln(3)+ln(4))

x=(ln(4)−ln(243))/(ln(9)+ln(4))

x=ln(4/243)/ln(36)

Final Answer

x=(ln(4)−5*ln(3))/(2*ln(3)+ln(4))

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