solve for x:log2x+log2(x+2)=3?

Answer 1

Let log mean log to base 2 in the following:-
log 2 + log(x + 2) = 3
log [ 2 (x + 2) ] = 3
2(x + 2) = 2 ³
2x + 4 = 8
2x = 4
x = 2

Answer 2

There are various levels right here. first of all this is less complicated to paintings in organic bases so word that if log_2(x) = y then x = 2^y = e^(ln(2^y)) (because of the fact e^(ln(w)) = w) = e^(y*ln(2)) (because of the fact ln(x^a) = a*ln(x).) So ln(x) = y*ln(2) or y = log_2(x) = ln(x)/ln(2). additionally, log(a/b) = log(a) - log(b), so log_2(x/4) = log_2(x) - log_2(4) = ln(x)/ln(2) - 2. So this will become 2*(ln(x)/ln(2))^2 - 3*(ln(x)/ln(2) - 2) - 11 = 0. Rearrange somewhat: 2*(ln(x)/ln(2))^2 - 3*(ln(x)/ln(2)) - 5 = 0 This looks like a quadratic with the x replaced via ln(x)/ln(2) - so returned enable y = ln(x)/ln(2), and then: 2*y^2 - 3*y - 5 = 0. remedy with the quadratic formula: a = 2, b = -3, c = -5; y = (-b (+-) sqrt(b^2 - 4ac))/2a = (3 (+-) sqrt(9 + 40))/(2*2) = (3 (+-) sqrt(40 9))/4 = 3/4 (+-) 7/4 so y = 10/4 = 5/2 or y = 3/4 - 7/4 = -4/4 = -a million. So if y = ln(x)/ln(2) = log_2(x), If log_2(x) = 5/2 then x = 2^(5/2) = 5.567. If log_2(x) = -2 then x = 2^(-a million) = 0.5. extra IN EDIT M5 - please be conscious the asker stated: "The bases are 2 and a couple of and the logarithmic numbers are x and x/4". this means your transformation of log(2x) (which i'm assuming from the algebra means 'log base 2 of 2x) is misguided - you have study the 1st term of the equation (2*log(2X)^2) as "2 circumstances the backside-2 logarithm of 2x"; yet because of the fact the asker stated that 'the logarithm numbers are x and x/4' it of course reads "2 circumstances the sq. of the backside-2 logarithm of x". in case you wade by your answer and altered lo(2x) with log(x) (in base 2 of direction), you detect your 3log(8) substitute to 3log(4) and get better the remarkable answer.

Answer 3

1 is the answer