Solve the equation log10 (5x+6)=2log10(5x-6)?

To solve the equation log10(5x+6) = 2log10(5x-6), we can use the properties of logarithms to simplify the equation.

First, we can use the property that log(a) = n is equivalent to 10^n = a. Applying this property to the equation, we get:

10^(log10(5x+6)) = 10^(2log10(5x-6))
5x+6 = (5x-6)^2

Now, we can expand the right side of the equation:

5x+6 = 25x^2 - 60x + 36

Rearranging the terms, we get:

25x^2 - 65x + 30 = 0

Now, we have a quadratic equation that we can solve using the quadratic formula:

x = (-(-65) ± √((-65)^2 - 4*25*30)) / 2*25
x = (65 ± √(4225 - 3000)) / 50
x = (65 ± √1225) / 50
x = (65 ± 35) / 50

Therefore, the solutions to the equation log10(5x+6) = 2log10(5x-6) are x = 2 and x = 3/5.

log10 (5x 6)=2log10(5x-6)

Using the logarithmic property log a^b = b log a

log10 (5x 6)= log10(5x-6)^2

Therefore,
(5x 6)=(5x-6)^2

5x 6= 25x^2 -60x +36

25x^2 -60x +36 -5x+6=0

25x^2 -65x +42=0

(5x -7)(5x -6)=0

5x-7=0 or 5x-6=0

5x=7 or 5x =6

x=7/5 or x=6/5

So the solutions are x=7/5 or x=6/5