Solve for x 7^(x/2)=5^(1-x)

Problem

7(x/2)=5(1−x)

Solution

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

ln(7(x/2))=ln(5(1−x))

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

x/2*ln(7)=(1−x)*ln(5)

3. Distribute the term on the right side of the equation.

x/2*ln(7)=ln(5)−x*ln(5)

4. Isolate the terms containing x on one side of the equation by adding x*ln(5) to both sides.

x/2*ln(7)+x*ln(5)=ln(5)

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

x*(ln(7)/2+ln(5))=ln(5)

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

x=ln(5)/(ln(7)/2+ln(5))

7. Simplify the denominator by multiplying the numerator and denominator by 2 to clear the fraction.

x=(2*ln(5))/(ln(7)+2*ln(5))

8. Apply logarithm properties to condense the expression, using n*ln(a)=ln(an) and ln(a)+ln(b)=ln(a*b)

x=ln(25)/(ln(7)+ln(25))

x=ln(25)/ln(175)

Final Answer

x=ln(25)/ln(175)

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