Hard Logarithm Problems With Solutions Pdf Upd -

(x = 2^\sqrt2) and (x = 2^-\sqrt2).

Find the integer (n) such that the equation [ \log_2 (x-1) + \log_2 (x-2) = \log_2 n ] has exactly one real solution. hard logarithm problems with solutions pdf

[ \sum_n=2^\infty \log_2 \left(1 + \frac1n\right)^\log_n 2 ] (x = 2^\sqrt2) and (x = 2^-\sqrt2)

Domain: (x>0, x\neq 1) and (x^2 - 5x+6 >0 \implies (x-2)(x-3)>0 \implies x<2) or (x>3) (intersect with (x>0) gives (0<x<2) or (x>3), and (x\neq1)). Log eq: (x^2 - 5x + 6 = x^1 \implies x^2 - 6x + 6 = 0 \implies x = 3 \pm \sqrt3). Check domain: (3+\sqrt3 \approx 4.73) OK. (3-\sqrt3 \approx 1.268) – is it <2? Yes, but not equal to 1, so OK. So both valid? Need (x\neq 1), (x>0). (3-\sqrt3 \approx 1.268) — is it in ((0,2))? Yes. So both work. Log eq: (x^2 - 5x + 6 =