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

Problem

5(x−2)=3(3*x+2)

Solution

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

ln(5(x−2))=ln(3(3*x+2))

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

(x−2)*ln(5)=(3*x+2)*ln(3)

3. Distribute the logarithmic constants into the terms inside the parentheses.

x*ln(5)−2*ln(5)=3*x*ln(3)+2*ln(3)

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

x*ln(5)−3*x*ln(3)=2*ln(3)+2*ln(5)

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

x*(ln(5)−3*ln(3))=2*ln(3)+2*ln(5)

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

x=(2*ln(3)+2*ln(5))/(ln(5)−3*ln(3))

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

x=(ln(9)+ln(25))/(ln(5)−ln(27))

x=ln(225)/ln(5/27)

Final Answer

x=(2*ln(3)+2*ln(5))/(ln(5)−3*ln(3))

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