Solve for x 4^(x+2)=6^(-2x-3)

Problem

4(x+2)=6(−2*x−3)

Solution

1. Apply the natural logarithm to both sides of the equation to bring the exponents down.

ln(4(x+2))=ln(6(−2*x−3))

2. Use the power rule of logarithms, ln(ab)=b*ln(a) to move the exponents to the front as coefficients.

(x+2)*ln(4)=(−2*x−3)*ln(6)

3. Distribute the logarithmic terms into the parentheses.

x*ln(4)+2*ln(4)=−2*x*ln(6)−3*ln(6)

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

x*ln(4)+2*x*ln(6)=−3*ln(6)−2*ln(4)

5. Factor out the common variable x from the left side.

x*(ln(4)+2*ln(6))=−3*ln(6)−2*ln(4)

6. Isolate x by dividing both sides by the expression in the parentheses.

x=(−3*ln(6)−2*ln(4))/(ln(4)+2*ln(6))

7. Simplify the expression using logarithm properties, specifically n*ln(a)=ln(an) and ln(a)+ln(b)=ln(a*b)

x=(−ln(6)−ln(4))/(ln(4)+ln(6))

x=(−(ln(216)+ln(16)))/(ln(4)+ln(36))

x=(−ln(3456))/ln(144)

Final Answer

x=(−3*ln(6)−2*ln(4))/(ln(4)+2*ln(6))

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