What is x if log2x = log5 + log(x - 24/5) ?

First, we'll impose the constraints of existence of
logarithms:

x - 24/5>0

x
>
24/5

2x>0

x>0

The
common interval of admissible values for x is (24/5
,+inf.).

Now,we'll could solve the equation using the property of
quotient:

log 2x = log 5 - log (x -
24/5)

log 2x  = log [5/(x -
24/5)]

Because the bases of logarithms are matching, we'll apply the
one to one property:

2x = 5/(x -
24/5)

We'll cross multiply;

2x(x -
24/5) = 5

We'll remove the
brackets:

2x^2 - 48x/5 - 5 = 0

We'll
multiply by 5:

10x^2 - 48x - 25 =
0

We'll apply the quadratic formula:

x1
= [48+sqrt(2304+1000)]/40

x1 = 2(24+sqrt826)/40 = 2.637 <
24/5

x2 = 2(24-sqrt826)/40 = -0.2<
24/5

Since the values of x1 and x2 do not belong
to the interval of admissible values, the equation has no valid
solutions