SOLUTION: Solve for x: Log2(11-6x) = 2Log2(x-1)+3

Question 554012: Solve for x:
Log2(11-6x) = 2Log2(x-1)+3
Found 2 solutions by rapaljer, Theo:
Answer by rapaljer(4671) (Show Source): You can put this solution on YOUR website!

First get all the log terms on one side:

By the Law of Logarithms:

By another Law of Logarithms, combine into a single log:

By the Definition of Logarithms:

Clear the fraction, by multiplying both sides by (x-1)^2:
8(x-1)^2= 11-6x
8(x^2-2x+1) =11-6x
8x^2-16x+8+6x-11=0
8x^2-10x-3=0

What are the chances that, after ALL this work, the trinomial will factor?? They usually do, in this type of problem!!

(4x +1)(2x-3)=0

There are two solutions: x=-1/4 and x=3/2

Of course, you can't have a log of a negative number, so x=-1/4 must be rejected. The other answer x=3/2 is acceptable.

You may want to see my FREE website for additional explanation on LOGARITHMS. The easiest way to find the website is to use the easy-to-remember and easy-to-spell link www.mathinlivingcolor.com. Near the bottom of this page is a link that takes you to my Homepage. On my Homepage, look for the link "Basic, Intermediate, and College Algebra: One Step at a Time."

Choose "College Algebra" and look in Chapter 4 for the sections on Logarithms. I think you will really like the "Math in Living Color" pages that go with these sections. In addition, I have 2 FREE videos on this topic that was made in my younger years before I retired. To see the videos, look on my Homepage for the link "Rapalje Videos in Living Color." Choose College Algebra, and look for the videos on "Logarithms."

If you need to contact me, send me an Email at rapaljer@seminolestate.edu.

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
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.

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