What is x if log(6) ( x+41 ) - log(6) (x+1) = 2 ?

log6 (x+ 41) - log6 (x+1) =
2

First we will find the
domain.

==> x+ 41>0 ==>  x >
-41

==> x + 1 > 0 ==> x >
-1

==> Then the domain is x > -1
.....(1) 

We will use the logarithm properties to solve for
x.

We know that log a - log b= log
(a/b)

==> log6 (x+41) / (x+1)
=2

Now we will rewrite using the exponent
form.

==> (x+41)/(x+1) =
6^2

==> (x+41)/ (x+1) =
36

Multiply by x+1

==>
x+ 41 = 36(x+1)

==> x+ 41 = 36x +
36

==> 35x =
5

=> x = 5/35 =
1/7

==> x= 1/7 > -1 ( then the solution
within the domain.)

Then the solution is x=
1/7