Calculate x if log2x=log5+log(x-24/5)

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We'll establish the constraints of existence of logarithms:

x - 24/5>0

x > 24/5

2x>0

x>0

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

We'll apply quotient property of logarithms:

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

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

Since 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 are not in the interval of admissible values, the equation has no solutions.

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