Solve log4 x + log4(x-2)=log4(15)

I know how do the example in my book but I don't know what I'm doing wrong here.

log4 x + log4(x-2)=log4(15)
log4 [x + log4(x-2)]=log4(15)
(x^2 + x)=4
x^2 + x-4=0

I see two major errors

1. from log A + log B = log (A x B)
your second line should have been
log4 [x(x-2)]=log4(15)

then in your third line

the right side should have been 15, not 4

so

x(x-2) = 15
x^2 - 2x - 15 = 0
(x-5)(x+2) = 0
x = 5 or x = -2, but x+-2 does not work since you cannot take the log of a negative

so x=5

should have been

x-5)(x+3) = 0
x = 5 or x = -3, but x=-3 does not work since you cannot take the log of a negative

so x =l 5 is still the only solution

To solve the equation, you can use the quadratic formula.

The quadratic equation in this case is:

x^2 + x - 4 = 0

Using the quadratic formula, which is given by:

x = (-b ± √(b^2 - 4ac)) / 2a

where a = 1, b = 1, and c = -4, we can substitute these values into the formula:

x = (-1 ± √(1^2 - 4(1)(-4))) / 2(1)

Simplifying further:

x = (-1 ± √(1 + 16)) / 2
x = (-1 ± √17) / 2

Therefore, the solutions for x are:

x = (-1 + √17) / 2
x = (-1 - √17) / 2

To solve the equation log4 x + log4(x-2) = log4 15, you can use logarithmic properties, specifically the product property of logarithms.

1. Start by combining the two logarithms on the left-hand side of the equation using the product property. According to this property, the sum of two logarithms with the same base is equal to the logarithm of their product:
log4 x + log4(x-2) = log4 (x * (x-2))

2. Simplify the equation:
log4 (x * (x-2)) = log4 15

3. Since the bases of the logarithms on both sides are the same (base 4), we can remove the logarithm from both sides of the equation:
x * (x-2) = 15

4. Expand the equation:
x^2 - 2x = 15

5. Rearrange the terms to form a quadratic equation in standard form:
x^2 - 2x - 15 = 0

Now you can solve this quadratic equation using factoring, completing the square, or using the quadratic formula. In this case, let's use factoring:

6. Factor the quadratic equation:
(x - 5)(x + 3) = 0

7. Set each factor equal to zero and solve for x:
x - 5 = 0 --> x = 5
x + 3 = 0 --> x = -3

Thus, the solution to the equation log4 x + log4(x-2) = log4 15 is x = 5 or x = -3.