Solve for x: lg(8x+9) + lgx = 1 + lg(x^2 - 1)

Given the logarithm
equation:

lg (8x+9) + lg (x) = 1+ lg
(x^2-1)

We need to find x
value.

We will use logarithm properties to
solve.

First, we know that lg a + lg b = lg
(ab)

==> lg (x(8x+9) = 1 + lg (x^2
-1)

Also, we know that lg 10 =
1

==> lg (8x^2 + 9x) = lg 10 + lg
(x^2-1)

==> lg (8x^2 +9x) = lg
10(x^2-1)

==> lg (8x^2 +9x) = lg (10x^2
-10)

Now we have the logs are equal, then the bases are
equal too.

==> 8x^2 + 9x = 10x^2 -
10

We will combine like
terms.

==> 2x^2 - 9x -10 =
0

Now we will find the
roots.

==> x1= ( 9 + sqrt(81+80) / 4 = (9+sqrt(161)
/ 4

==> x2= (9-sqrt(161) / 4 ( Not
valid)

Then, the answer
is:

x = (9 + sqrt161)
/4