Find the Null Space ( log of 2x=1+ log of y-2)

Problem

(log_)(2*x)=1+(log_)(y−2)

Solution

1. Identify the task as finding the set of all points (x,y) that satisfy the given logarithmic equation, assuming the standard base 10 for (log_)()

2. Apply the definition of the constant 1 by rewriting it as a logarithm with the same base, so 1=(log_)(10)

3. Use the product rule for logarithms, (log_)(a)+(log_)(b)=(log_)(a*b) to combine the terms on the right side of the equation.

(log_)(2*x)=(log_)(10)+(log_)(y−2)

(log_)(2*x)=(log_)(10*(y−2))

4. Remove the logarithms by applying the property that if (log_)(a)=(log_)(b) then a=b

2*x=10*(y−2)

5. Simplify the equation by dividing both sides by 2.

x=5*(y−2)

x=5*y−10

6. Determine the domain constraints for the logarithms to be defined, which require the arguments to be positive: 2*x>0 and y−2>0

x>0

y>2

Final Answer

{(x,y)∈ℝ2:x=5*y−10,y>2}

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