solve logarithmic equation in exact form only. show work.

log_5(x-3)+ log_5(x-6)-log_5(x+1)=2

log_5 [(x-3)(x-6)/(x+1) = 2 , we know x > 6

(x-3)(x-6)/(x+1) = 5^2
x^2 - 9x + 18 = 25x + 25
x^2 - 34x - 7 = 0

x = (34 ± √1184)/2
= (34 ± 4√74)/2
= 17 ± 2√74 , but x > 6

x = 17 + 2√74

Thanks Reiny - you have rescued me again

To solve the logarithmic equation log_5(x-3) + log_5(x-6) - log_5(x+1) = 2, we need to use logarithmic properties. One property that will be useful here is the product property, which states that the logarithm of a product is equal to the sum of the logarithms of its factors.

First, let's apply the product property to simplify the equation:

log_5((x-3)(x-6)) - log_5(x+1) = 2

Next, we can apply another logarithmic property, which is the quotient property. This property states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator.

log_5(((x-3)(x-6))/(x+1)) = 2

Now, we can convert the equation to exponential form using the definition of logarithm. In exponential form, the base of the logarithm is raised to the power equal to the value of the logarithm.

5^2 = ((x-3)(x-6))/(x+1)

Simplifying this equation further:

25 = (x^2 - 9x +18)/(x+1)

Multiplying both sides of the equation by (x+1):

25(x+1) = x^2 - 9x + 18

Expanding the equations on both sides:

25x + 25 = x^2 - 9x + 18

Rearranging the terms:

x^2 - 34x + 7 = 0

Now, we need to solve this quadratic equation. One approach to solving this equation is by factoring. However, this equation cannot be easily factored. We can use the quadratic formula instead:

x = (-b ± √(b^2 - 4ac))/(2a)

In this case, a = 1, b = -34, and c = 7. Substituting these values into the quadratic formula:

x = (-(-34) ± √((-34)^2 - 4(1)(7))) / (2(1))

Simplifying this equation further:

x = (34 ± √(1156 - 28)) / 2

x = (34 ± √1128) / 2

Now, we need to calculate the square root of 1128:

x = (34 ± 33.61) / 2

This gives us two possible solutions:

x1 = (34 + 33.61) / 2 ≈ 33.805
x2 = (34 - 33.61) / 2 ≈ 0.195

Therefore, the exact solutions to the logarithmic equation log_5(x-3) + log_5(x-6) - log_5(x+1) = 2 are x1 ≈ 33.805 and x2 ≈ 0.195.