Question 554012
you have:
log(2,11-6x) = 2log(2,x-1)+3
subtract 2log(2,x-1) from both sides of this equation to get:
log(2,11-6x) - 2log(2,x-1) = 3
since a*log(b) = log(b^a), your equation becomes:
log(2,11-6x) - log(2,(x-1)^2) = 3
since log(a) - log(b) = log(a/b), your equation becomes:
log(2,(11-6x)/(x-1)^2) = 3
since log(2,a) = b if and only if 2^b = a, your equation becomes:
2^3 = (11-6x)/(x-1)^2
simplify to get:
8 = (11-6x)/(x-1)^2
multiply both sides of this equation by (x-1)^2 to get:
8*(x-1)^2 = 11-6x
simplify to get:
8*(x^2 - 2x + 1) = 11 - 6x
simplify further to get:
8x^2 - 16x + 8 = 11 - 6x
subtract 11 and add 6x to both sides of this equation to get:
8x^2 - 16x + 6x + 8 - 11 = 0
combine like terms to get:
8x^2 - 10x - 3 = 0
factor this to get:
(4x+1)(2x-3) = 0
solve for x to get:
x = -1/4 or x = 3/2
substitute in your original equations to see if these answers check out.
x = -1/4 doesn't check out because it leads to a log of a negative number which is not allowed.
x = 3/2 does check out, so x = 3/2 is your answer.
your original equation is:
log(2,11-6x) = 2log(2,x-1)+3
when x = 3/2, this equation becomes:
log(2,2) = 2log(2,1/2) + 3
we use the base conversion formula to make these into equivalent logs to the base of 10 which we can solve using our calculator.
log(2,x) = log(10,x) / log(10,2)
we get:
log(2,2) = 2log(2,1/2) + 3 becomes:
log(10,2)/log(10,2) = 2log(10,1/2)/log(10,2) + 3
this becomes:
.30103000/.30103000 = 2*-.30103000/.30103000 + 3 which becomes:
1 = 2*-1 + 3 which becomes:
1 = 1
this confirms the value of x = 3/2 is good.