Solve for x log 3 (2x+3) -1 = 0.

We'll start by imposing the constraints of existance of
logarithm
function.

2x+3>0

We'll
add -3 both
sides:

2x>-3

We'll
divide by
2:

x>-3/2

So, for the
logarithms to exist, the values of x have to belong to the interval (-3/2,
+inf.)

We'll shift the free term to the right
side:

 log 3 (2x+3) = 1

We'll
create matching bases to the right side.

 log 3 (2x+3)
=  log 3 (3)

Now, because the bases are
matching, we'll apply the one to one
property:

2x+3 = 3

We'll
eliminate like terms:

2x =
3-3

2x = 0

We'll divide by
2:

x = 0 >
-3/2

Since the value for x belongs to the
interval (-3/2,+inf.), the solution is valid.