f(2) = 0 so 2 is the correct answer. You made a small typo which I've marked in red.

x = 2 is a solution but x= -4 is not a solution. This is because you cannot take the log of a negative number or 0. In other words x > 02. log2(x) + log2(x + 2) = 3

log2((x)(x + 2)) = 3

log2(x(x + 2)) = 3

(x(x + 2)) = 2^3

(x(x + 2)) = 8

x(x+2) = 8

(x(x) + x(2)) = 8

(x^2 + 2x) = 8

x^2 + 2x = 8

x^2 + 2x - 8 = 0

(x + 4)(x - 2) = 0

x + 4 = 0

x - 2 = 0

x = -4

x = 2

x = -4, 2

I've made two omissions at the start of lines three and 4 because they are not necessary.

Correct apart from the x I removed in line 4 (you took away 2 not 2x so I'm guessing it was a typo)3. 2^(2x + 2) = 64

2^(2x + 2) = 2^6

(2x + 2) = 6

2x + 2 = 6

2x = -2 + 6

2x = 4

2x/2 = 4/2

x = 4/2

x = 2

Correct4. 3^(x^2 + 2x) = 27

3^(x^2 + 2x) = 3^3

(x^2 + 2x) = 3

x^2 + 2x = 3

x^2 + 2x - 3 = 0

(x + 3)(x-1) = 0

x + 3 = 0

x - 1 = 0

x = -3

x = 1

x = -3, 1

Correct.5. 4^x = 63

ln(4^x) = ln(63)

xln(4) = ln(63)

xln(4)/ln(4) = ln(63)/ln(4)

x = ln(63)/ln(4)

Correct6. 10^4x = 101

ln(10^4x) = ln(101)

4xln(10) = ln(101)

4xln(10)/4ln(10) = ln(101)/4ln(10)

x = ln(101)/4ln(10)