Determine the solutions of the equation lg(x+1) - lg9 = 1 - lg(x)

Let's use some logarithmic rules
before:

1=lg10

lgx-lgy=lg(x/y)

x>0

Now,
let's solve the
equation:

lg(x+1)-lg9=lg((x+1)/9)

1-lgx=lg10-lgx=lg(10/x)

lg((x+1)/9)=lg(10/x)

From
one of the properties of logarithmic function, the one which says that this function is
an
injection:

(x+1)/9=10/x

We'll
use the cross
multiplying:

x*(x+1)=9*10

x^2
+x -90=0

x1=[-1+ sq
root(1+4*90)]/2=(-1+19)/2=9

x2=(-1-19)/2=-10

From
the existence condition of the logarithm, x>0, so the only accepted solution is
x1=9