SOLUTION: solve: log subscript 4(10-x)=2-log subscript 4(-x-5). Thanks!

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: solve: log subscript 4(10-x)=2-log subscript 4(-x-5). Thanks!      Log On


   

Question 246654: solve:
log subscript 4(10-x)=2-log subscript 4(-x-5).
Thanks!
Answer by rapaljer(4671) About Me  (Show Source):
You can put this solution on YOUR website!
log%284%2C10-x%29=2-log%284%2C-5-x%29

Start by getting the logs on the same side of the equation:
log%284%2C10-x%29%2B+log%284%2C-5-x%29=2

By the Law of Logarithms, you can write this:
log%284%2C%2810-x%29%28-5-x%29%29=2
log%284%2C-50-5x%2Bx%5E2%29=2

By the definition of logarithms:
4%5E2=+x%5E2-5x-50%29
16=x%5E2+-5x-50
0=x%5E2-5x-66
0=%28x-11%29%28x%2B6%29

There are TWO solutions to this equation:
x=11 and x=-6

HOWEVER, you MUST CHECK to make sure you didn't "accidentally" cause any logs of negatives--these are NOT ALLOWED!

If x= 11, in the original equation, you get TWO expressions in which you have a log of a negative number. This answer must be rejected!

If x=-6, both logarithms in the original equation come out okay, so this is the only answer that works. YOU MUST REJECT x=11.

For additional help with LOGARITHMS, one of my favorite topics, you really need to see my website, where I have a LOT of resources to help you. To find my website, click on my tutor name "Rapaljer" anywhere in algebra.com. This takes you to a page that gives a link to my website. Click on this link, and from the Homepage of my website, look for "Basic, Intermediate, and College Algebra: One Step at a Time." Choose "College Algebra", and in "Chapter 4" you will find my entire chapter on LOGARITHMS. I have my own explanation, especially written for students who have trouble with math! My students unanimously have told me over the years that they find my explanations, examples, and exercises MUCH easier to understand than traditional textbooks. Many of the exercises from these pages are solved in COLOR on my "MATH IN LIVING COLOR" pages that go with my curriculum.

In addition to this written explanation, I have TWO 2-hour videos of me teaching the entire subject of LOGARITHMS in my own classes a few (quite a few--I was a LOT younger then, but weren't we all!!) years ago. To see my videos, ALL FREE,from my Homepage, look for "Rapalje Videos in Living Color." Then click on "College Algebra", and look for "Logarithms Part I" and "Logarithms Part II." This particular question would probably be explained in Logarithms Part II. I hope you will try my website, try my curriculum, and try my videos! They are all FREE! If you like them, then please recommend them to your friends and family.

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus