# Solve the logarithmic equation explaining the properties used log 6 (3x+14)-log 6 (5) = log 6 (2x)

hala718 | High School Teacher | (Level 1) Educator Emeritus

Posted on

log 6(3x+14)-log 6(5) = log 6(2x)

==> log 6[(3x+14)/5] = log 6(2x)

Now since log x = log y ==> x=y

==> 6(3x+14)/5 = 6(2x)

==> 3x + 14=  (5)(2) x

==> 3x +14 = 10 x

==> x = 14/7= 2

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

First, we have to use the quotient property of the logarithms:

log 6 [(3x+14)/5] = log 6 (2x)

Now, we'll have to use the one to one property, that means that:

log 6 [(3x+14)/5] = log 6 (2x) if and only if (3x+14)/5=2x

After cross multiplying, we'll get:

3x+14=10x

We'll move the terms to one side:

10x-3x=14

7x=14

x=14/7

x=2

If we'll check the solution into equation, we'll get:

log 6 [(3*2+14)/5] = log 6 (2*2)

log 6 [(20)/5] = log 6 (4)

log 6 (4) = log 6 (4)

neela | High School Teacher | (Level 3) Valedictorian

Posted on

To solve log6(3x+14)-log6(5) = log6(2x).

Solution:

log6(3x+14)-log6(5) = log6(2x). Or

log6{3x+14)-log6(5)- log6(2x) = 0. Or

log6{(3x+14)/(5*2x)} = 0. Tkaing anti log with respect to 6,

(3x+14)/(10x) = 6^0 = 1. Or

3x+14 = 10x. Or

14  = 10x-7x. Or

14 = 7x . Or

2 = x. Or  x = 2.