Solve for x log of 8x- log of 1+ square root of x=2

Problem

(log_)(8*x)−(log_)(1+√(,x))=2

Solution

1. Apply the quotient rule for logarithms, which states that (log_)(a)−(log_)(b)=(log_)(a/b)

(log_)((8*x)/(1+√(,x)))=2

2. Rewrite in exponential form by assuming the base is 10 (common logarithm).

(8*x)/(1+√(,x))=10^2

3. Simplify the constant on the right side of the equation.

(8*x)/(1+√(,x))=100

4. Multiply both sides by the denominator to clear the fraction.

8*x=100*(1+√(,x))

5. Distribute and simplify the equation by dividing both sides by 4

2*x=25*(1+√(,x))

2*x=25+25√(,x)

6. Rearrange into a quadratic form by letting u=√(,x) which implies u^2=x

2*u^2−25*u−25=0

7. Apply the quadratic formula u=(−b±√(,b^2−4*a*c))/(2*a) to solve for u

u=(25±√(,(−25)^2−4*(2)*(−25)))/(2*(2))

u=(25±√(,625+200))/4

u=(25±√(,825))/4

8. Simplify the radical √(,825)=√(,25⋅33)=5√(,33)

u=(25±5√(,33))/4

9. Solve for x by squaring u noting that u=√(,x) must be positive, so we discard the negative root.

x=((25+5√(,33))/4)^2

10. Expand the square to find the final value of x

x=(625+250√(,33)+25*(33))/16

x=(625+825+250√(,33))/16

x=(1450+250√(,33))/16

x=(725+125√(,33))/8

Final Answer

(log_)(8*x)−(log_)(1+√(,x))=2⇒x=(725+125√(,33))/8

Want more problems? Check here!