Question

Solve the equation: 4 log 3 × log x
/ log 9 = log 27​

Answer 1

Answer:

Solve for x:

(log(27) x^2)/log(9) = x + 4

　

Subtract x + 4 from both sides:

(log(27) x^2)/log(9) - x - 4 = 0

　

Divide both sides by log(27)/log(9):

-(log(9) x)/log(27) + x^2 - (4 log(9))/log(27) = 0

　

Add (4 log(9))/log(27) to both sides:

x^2 - (log(9) x)/log(27) = (4 log(9))/log(27)

　

Add (log^2(9))/(4 log^2(27)) to both sides:

-(log(9) x)/log(27) + x^2 + (log^2(9))/(4 log^2(27)) = (log^2(9))/(4 log^2(27)) + (4 log(9))/log(27)

　

Write the left hand side as a square:

(x - log(9)/(2 log(27)))^2 = (log^2(9))/(4 log^2(27)) + (4 log(9))/log(27)

　

Take the square root of both sides:

x - log(9)/(2 log(27)) = sqrt((log^2(9))/(4 log^2(27)) + (4 log(9))/log(27)) or x - log(9)/(2 log(27)) = -sqrt((log^2(9))/(4 log^2(27)) + (4 log(9))/log(27))

　

Add log(9)/(2 log(27)) to both sides:

x = sqrt((log^2(9))/(4 log^2(27)) + (4 log(9))/log(27)) + log(9)/(2 log(27)) or x - log(9)/(2 log(27)) = -sqrt((log^2(9))/(4 log^2(27)) + (4 log(9))/log(27))

Add log(9)/(2 log(27)) to both sides:

| x = sqrt((log^2(9))/(4 log^2(27)) + (4 log(9))/log(27)) + log(9)/(2 log(27)) or x = log(9)/(2 log(27)) - sqrt((log^2(9))/(4 log^2(27)) + (4 log(9))/log(27))

x = -4/3 and x = 2

Domain: {x element R : 0 1} (assuming a function from reals to reals)