Solve for x: log2(x-4)+log2(x)=5?

log2(x-4)+log2(x)=5
x(x-4)=2^5
x^2-4x=32
x^2-4x-32=0
(x+4)(x-8)=0
x+4=0 x-8=0
x=-4 x=8 answer//

There are a number of tiers right here. at first this is greater basic to artwork in organic bases so word that if log_2(x) = y then x = 2^y = e^(ln(2^y)) (considering e^(ln(w)) = w) = e^(y*ln(2)) (considering 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 slightly: 2*(ln(x)/ln(2))^2 - 3*(ln(x)/ln(2)) - 5 = 0 This appears like a quadratic with the x replaced by ln(x)/ln(2) - so back enable y = ln(x)/ln(2), and then: 2*y^2 - 3*y - 5 = 0. clean up 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 word the asker stated: "The bases are 2 and a pair of and the logarithmic numbers are x and x/4". this suggests your transformation of log(2x) (which i'm assuming from the algebra ability 'log base 2 of 2x) is misguided - you have examine 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 for sure 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) exchange to 3log(4) and get well the wonderful answer.

log_2(x - 4) + log_2(x) = 5
log_2[x(x - 4)] = 5
x(x - 4) = 2^5
x^2 - 4x = 32
x^2 - 4x - 32 = 0
x^2 + 4x - 8x - 32 = 0
(x^2 + 4x) - (8x + 32) = 0
x(x + 4) - 8(x + 4) = 0
(x + 4)(x - 8) = 0

x + 4 = 0
x = -4

x - 8 = 0
x = 8 [log(-4) is undefined, so x just equals 8]

∴ x = 8

x(x-4)=2^5 = 32 --->(x-8)(x+4) = 0---->x = 8, -4
answer x = 8