Jawaban:

Input

3/2 log(9/4) + 1/4 log(1/16)

Exact result

3/2 log(9/4) - log(16)/4

Decimal approximation

0.5232481437645478365168072249348708416404711360272273387213629629...

Property

3/2 log(9/4) - log(16)/4 is a transcendental number

Alternate forms

log(27/16)

3 log(3) - 4 log(2)

1/4 (6 log(9/4) - log(16))

3/2 (2 log(3) - 2 log(2)) - log(2)

Number line

Number line

Continued fraction

[0; 1, 1, 10, 3, 1, 16, 1, 6, 1, 4, 4, 3, 1, 9, 1, 2, 297, 5, 14, 12, 1, 11, 1, 46, 1, 1, 6, ...]

Alternative representations

1/2 log(9/4) 3 + 1/4 log(1/16) = 3/2 log(a) log(a, 9/4) + 1/4 log(a) log(a, 1/16)

1/2 log(9/4) 3 + 1/4 log(1/16) = (3 log(e, 9/4))/2 + log(e, 1/16)/4

1/2 log(9/4) 3 + 1/4 log(1/16) = -3/2 Li_1(1 - 9/4) - 1/4 Li_1(1 - 1/16)

Series representations

1/2 log(9/4) 3 + 1/4 log(1/16) = 3/2 log(5/4) - log(15)/4 + sum_(k=1)^∞-((-1/15)^k (-1 + 2^(1 + 2 k) 3^(1 + k)))/(4 k)

1/2 log(9/4) 3 + 1/4 log(1/16) = 5/2 i π floor((π - arg(1/z_0) - arg(z_0))/(2 π)) + (5 log(z_0))/4 + sum_(k=1)^∞-((-1)^k (6 (9/4 - z_0)^k - (16 - z_0)^k) z_0^(-k))/(4 k)

1/2 log(9/4) 3 + 1/4 log(1/16) = 3 i π floor(arg(9/4 - x)/(2 π)) - 1/2 i π floor(arg(16 - x)/(2 π)) + (5 log(x))/4 + sum_(k=1)^∞-((-1)^k (6 (9/4 - x)^k - (16 - x)^k) x^(-k))/(4 k) for x<0

Integral representations

1/2 log(9/4) 3 + 1/4 log(1/16) = integral_1^(9/4) (3/(11 - 12 t) + 3/(2 t)) dt

1/2 log(9/4) 3 + 1/4 log(1/16) = integral_(-i ∞ + γ)^(i ∞ + γ)-(i 15^(-s) (-1 + 2^(1 + 2 s) 3^(1 + s)) Γ(-s)^2 Γ(1 + s))/(8 π Γ(1 - s)) ds for -1<γ<0