Solve for xlg(x+1)-lg9=1-lgx
2 Answers
justaguide | College Teacher | (Level 2) Distinguished Educator
The equation given is lg(x+1) - lg 9 = 1 - lg x
I assume the base of the logarithm is 10. Use the property that lg a - lg b = lg a/b and 1 = lg 10
lg(x+1) - lg 9 = 1 - lg x
=> lg(x+1) - lg 9 = lg 10 - lg x
=> lg [(x+1)/9] = lg (10/x)
(x + 1)/9 = 10/x
=> x^2 + x = 90
=> x^2 + 10x - 9x -90 = 0
=> x(x + 10) - 9(x + 10) = 0
=> (x -9)(x + 10) = 0
x = 9 and x = -10
As log of negative numbers is not defined we eliminate x = -10
The solution of the equation is x = 9
giorgiana1976 | College Teacher | (Level 3) Valedictorian
We'll re-write the equation, moving the terms in x to the left side and the terms without x, to the right side.
lg(x+1) - lg9 = 1 - lgx
lg(x+1) + lgx = 1 + lg9
We'll re-write 1 as lg 10
lg(x+1) + lgx = lg 10 + lg9
Since the bases are matching, we'll use the product rule of logarithms both sides:
lg x(x+1) = lg 90
Since the bases are matching, we'll use one to one rule:
x(x+1) = 90
We'll remove the brackets:
x^2 + x - 90 = 0
We'll apply the quadratic formula:
x1 = [-1+sqrt(1+360)]/2
x1 = (-1+19)/2
x1 = 9
x2 = (-1-19)/2
x2 = -10
We'll reject the second solution, because it's negative. We'll keep the solution x = 9.