Find the Domain 5x^(2y)=30

Problem

5*x(2*y)=30

Solution

1. Isolate the exponential term by dividing both sides of the equation by 5.

x(2*y)=6

2. Apply the natural logarithm to both sides to bring the variables out of the exponent.

ln(x(2*y))=ln(6)

3. Use the power rule of logarithms, ln(ab)=b*ln(a) to rewrite the left side.

2*y*ln(x)=ln(6)

4. Identify the restriction on the logarithmic function ln(x) which requires the argument to be strictly greater than zero.

x>0

5. Identify the restriction on the base of the exponential expression x(2*y) For the expression to be defined for all real values of 2*y the base x must be positive.

x>0

6. Check for values that would make the equation impossible. If x=1 the equation becomes 1(2*y)=6 which simplifies to 1 = 6.S*i*n*c*e*t*h*i*s(i)*s(a)*c*o*n*t*r*a*d(i)*c*t*i*o*n,$ cannot be 1.

x≠1

7. Combine the conditions to define the domain for x

x∈(0,1)∪(1,∞)

Final Answer

Domain:{[x∈ℝ,x>0,x≠1]}

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