SOLUTION: Find the exact solution, using common logarithms. log(x − 3) − log(5x − 7) = log(1/x)

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Find the exact solution, using common logarithms. log(x − 3) − log(5x − 7) = log(1/x)      Log On


   

Question 1147951: Find the exact solution, using common logarithms.
log(x − 3) − log(5x − 7) = log(1/x)
Found 3 solutions by josgarithmetic, greenestamps, ikleyn:
Answer by josgarithmetic(39618) About Me  (Show Source):
You can put this solution on YOUR website!
log%28%28%28x-3%29%2F%285x-7%29%29%29=log%28%281%2Fx%29%29
%28x-3%29%2F%285x-7%29=1%2Fx
5x-5=x%5E2-3x
x%5E2-8x%2B7=0
%28x-7%29%28x-1%29=0

system%28x=7%2Cor%2Ccross%28x=1%29%29--------








--------------
note: original equation was log%28%28x-3%29%29-log%28%285x-7%29%29=log%28%281%2Fx%29%29.

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


NO! The two solutions do NOT both work!

The given equation is

(1) log(x − 3) − log(5x − 7) = log(1/x)

To start solving that, you use rules of logarithms:

(2) log((x-3)/(5x-7)) = log(1/x)

Working from there, you find apparent solutions x=1 and x=7.

And those solutions both work in (2).

BUT... they have to satisfy the ORIGINAL equation. And x=1 does not, because it results in trying to take the logarithm of a negative number.

ANSWER: x=7 is the only solution.

Answer by ikleyn(52794) About Me  (Show Source):
You can put this solution on YOUR website!
.

Had you answer as @josgarithmetic did,  you immediately would get the score from  "1"  to  "2"  of  4  from your teacher.