SOLUTION: log(5x) = log3+log(x-2) ...Just like that; no bases. PLEASE HELP!! -If you can go step by step with explanations that would be sooo helpful. Thank you!!

Algebra ->  Matrices-and-determiminant -> SOLUTION: log(5x) = log3+log(x-2) ...Just like that; no bases. PLEASE HELP!! -If you can go step by step with explanations that would be sooo helpful. Thank you!!       Log On


   

Question 925847: log(5x) = log3+log(x-2) ...Just like that; no bases. PLEASE HELP!! -If you can go step by step with explanations that would be sooo helpful. Thank you!!
Found 3 solutions by lwsshak3, josmiceli, Edwin McCravy:
Answer by lwsshak3(11628) About Me  (Show Source):
You can put this solution on YOUR website!
log(5x) = log3+log(x-2)
when no base is shown, the base=10.
log%285x%29-log%28x-2%29=log%283%29
log%28%285x%2F%28x-2%29%29%29=log%28%283%29%29
5x%2F%28x-2%29=3
5x=3x-6
2x=-6
x=-3
no solution: x>0

Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
When the base is not shown, the base is
normally +10+, because there has to
be some base
----------------
+log%28+5x+%29+=+log%283%29+%2B+log%28%28+x-2+%29%29+
use the following general rule on the right side:
+log%28+a+%29+%2B+log%28+b+%29+=+log%28+a%2Ab+%29+
+log%28+5x+%29+=+log%28%28+3%2A%28+x+-+2+%29%29%29+
-----------------------------
The next step can be stated as:
If the logs of 2 expressions are
equal, then the result of raising
the bases to those logs are equal, so
---------------------------------
+5x+=+3%2A%28+x-2+%29+
+5x+=+3x+-+6+
+2x+=+-6+
+x+=+-3+
------------------
check the answer:
+log%28+5x+%29+=+log%283%29+%2B+log%28%28+x-2+%29%29+
+log%28%28+5%2A%28-3%29+%29%29+=+log%283%29+%2B+log%28%28+-3-2+%29%29+
+log%28%28+-15+%29%29+=+log%28%28+3%2A%28-5%29+%29%29+
+-15+=+-15+
-------------------
There is a problem with this, and you should get
more opinions on it. You can't possible get a
negative result by raising +10+ ( or any (+) base )
to any log. This seems to be what is happening.
What's the deep explanation, or is this a bogus
problem? Try to get an answer.



Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
log%28%285x%29%29+=+log%28%283%29%29%2Blog%28%28x-2%29%29

The base is 10 anytime the base is not written.
But actually it doesn't matter what the base is.
We just need to know some rules of logaritms

Rule 1:   log%28%28A%29%29%2Blog%28%28B%29%29=log%28%28A%2AB%29%29

Rule 2:  If log%28%28A%29%29=log%28%28B%29%29 then we can drop the logs and write A=B

Start with 

log%28%285x%29%29+=+log%28%283%29%29%2Blog%28%28x-2%29%29

We use rule 1 on the right side:

log%28%285x%29%29+=+log%28%283%28x-2%29%29%29

Now use rule 2

5x=3%28x-2%29

That's an easy equation to solve.  You should get -3

Trouble is, you must always check your answer to see if it
is an actual solution or just an extraneous answer.

log%28%285x%29%29+=+log%28%283%29%29%2Blog%28%28x-2%29%29

log%28%285%28-3%29%29%29+=+log%28%283%29%29%2Blog%28%28-3-2%29%29
log%28%28-15%29%29+=+log%28%283%29%29%2Blog%28%28-5%29%29

Oh oh!  -3 is an extraneous answer because there is no such
number in real number mathematics as the logarithm of a negative
number.  Since when we checked we got a logarithm of a negative
number, we knew that the answer we got was extraneous and not a
real solution.

THERE IS NO SOLUTION!

Edwin