Question 554012
{{{log(2,(11-6x)) = 2log(2,(x-1)) + 3}}}


First get all the log terms on one side:
{{{log(2,(11-6x))-2log(2,(x-1)) = 3}}}


By the Law of Logarithms:
{{{log(2,(11-6x)) -log(2,(x-1)^2) = 3}}}


By another Law of Logarithms, combine into a single log:
{{{log(2,((11-6x)/(x-1)^2)) = 3}}}


By the Definition of Logarithms:
{{{2^3= (11-6x)/(x-1)^2}}}
{{{8 = (11-6x)/(x-1)^2}}}


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