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Great!!! Thanks!!!!
mesmerizing.
❤😮😊
Very good!
Thank you, Cap!
Good @MATHS TUTORIAL (BL SAHU)
❤
Method is unnecessarily complicated.4^2a=48Take ln both sides:2a*ln4=ln(4*4*3)2a*ln4=2ln4+ln3Divide both sides by 2ln4a=1+ln3)/2ln4But ln4=2ln2a=1+ln3/4ln2
Entirely agree.
Including complex roots: 4^a * 4^a = 484^(a + a) = (48^[1 / 2])^24^(2 * a) = ([2^4 * 3]^[1 / 2])^24^(a * 2) = ([2^4]^[1 / 2] * 3^[1 / 2])^2(4^a)^2 = (2^[4 * (1 / 2)] * 3^[1 / 2])^2(4^a)^2 = (2^2 * 3^[1 / 2])^2(4^a)^2 = (4 * 3^[1 / 2])^2sqrt([4^a]^2) = +/- sqrt([4 * 3^(1 / 2)]^2)4^a = +/- 4 * 3^(1 / 2)(1 / 4) * 4^a = +/- (1 / 4) * 4 * 3^(1 / 2)4^a * 4^(-1) = +/- 4^1 * 4^(-1) * 3^(1 / 2)4^(a - 1) = +/- 4^(1 - 1) * 3^(1 / 2)4^(a - 1) = +/- 4^0 * 3^(1 / 2)4^(a - 1) = +/- 1 * 3^(1 / 2)4^(a - 1) = +/- 3^(1 / 2)(2^2)^(a - 1) = +/- ([3^(1 / 2)]^[1 / 2])^22^(2 * [a - 1]) = +/- (3^[(1 / 2) * (1 / 2)])^22^([a - 1] * 2) = +/- (3^[1 / 4])^2(2^[a - 1])^2 = +/- (3^[1 / 4])^2(2^[a - 1])^2 = +(3^[1 / 4])^2, or (2^[a - 1])^2 = -(3^[1 / 4])^2Let x = 2^(a - 1), and y = 3^(1 / 4)(2^[a - 1])^2 = (3^[1 / 4])^2, or (2^[a - 1])^2 = -(3^[1 / 4])^2=> x^2 = y^2, or x^2 = -y^2=> x^2 - y^2 = y^2 - y^2, or x^2 + y^2 = -y^2 + y^2=> x^2 - y^2 = 0, or x^2 + y^2 = 0=> (x - y)(x + y) = 0, or (x - i * y)(x + i * y) = 0=> (2^[a - 1] - 3^[1 / 4])(2^[a - 1] + 3^[1 / 4]) = 0, or (2^[a - 1] - i * 3^[1 / 4])(2^[a - 1] + i * 3^[1 / 4]) = 0Suppose 2^(a - 1) - 3^(1 / 4) = 02^(a - 1) - 3^(1 / 4) = 02^(a - 1) - 3^(1 / 4) + 3^(1 / 4) = 0 + 3^(1 / 4)2^(a - 1) = 3^(1 / 4)ln(2^[a - 1]) = ln(3^[1 / 4])(a - 1) * ln(2) = (1 / 4) * ln(3)(a - 1) * ln(2) / ln(2) = ln(3) / (4 * ln[2])(a - 1) * log_2(2) = log_2(3) / 4(a - 1) * 1 = log_2(3) / 4a - 1 + 1 = log_2(3) / 4 + 1a = log_2(3) / 4 + 1Suppose 2^(a - 1) + 3^(1 / 4) = 02^(a - 1) + 3^(1 / 4) - 3^(1 / 4) = 0 - 3^(1 / 4)2^(a - 1) = -3^(1 / 4)ln(2^[a - 1]) = ln(-1 * 3^[1 / 4])ln(2^[a - 1]) = ln(i^2 * 3^[1 / 4])(a - 1) * ln(2) = ln(i^2 * 3^[1 / 4])(a - 1) * ln(2) / ln(2) = ln(i^2 * 3^[1 / 4]) / ln(2)(a - 1) * log_2(2) = ln(i^2) / ln(2) + ln(3^[1 / 4]) / ln(2)(a - 1) * 1 = ln(e^[i * tau / 2]) / ln(2) + log_2(3^[1 / 4])a - 1 = (i * tau / 2) * ln(e) / ln(2) + log_2(3^[1 / 4])a - 1 = (i * tau / 2) * 1 / ln(2) + log_2(3^[1 / 4])a - 1 = i * tau / (2 * ln[2]) + (1 / 4) * log_2(3)a - 1 + 1 = i * tau / (2 * ln[2]) + log_2(3) / 4 + 1a = i * tau / (2 * ln[2]) + log_2(3) / 4 + 1a = i * 2 * tau / (2^2 * ln[2]) + log_2(3) / 4 + 1a = i * 2 * tau / (4 * ln[2]) + log_2(3) / 4 + 1Suppose 2^(a - 1) - i * 3^(1 / 4) = 02^(a - 1) - i * 3^(1 / 4) = 02^(a - 1) - i * 3^(1 / 4) + i * 3^(1 / 4) = 0 + i * 3^(1 / 4)2^(a - 1) = i * 3^(1 / 4)ln(2^[a - 1]) = ln(i * 3^[1 / 4])(a - 1) * ln(2) = ln(i * 3^[1 / 4])(a - 1) * ln(2) / ln(2) = ln(i * 3^[1 / 4]) / ln(2)(a - 1) * log_2(2) = ln(i) / ln(2) + ln(3^[1 / 4]) / ln(2)(a - 1) * 1 = ln(e^[i * tau / 4]) / ln(2) + (1 / 4) * ln(3^[1 / 4]) / ln(2)a - 1 = (i * tau / 4) * ln(e) / ln(2) + ln(3) / (4 * ln[2])a - 1 = (i * tau / 4) * 1 / ln(2) + ln(3) / (4 * ln[2])a - 1 = i * tau / (4 * ln[2]) + ln(3) / (4 * ln[2])a - 1 = i * tau / (4 * ln[2]) + log_2(3) / 4a - 1 + 1 = i * tau / (4 * ln[2]) + log_2(3) / 4 + 1a = i * 1 * tau / (4 * ln[2]) + log_2(3) / 4 + 1Suppose 2^(a - 1) + i * 3^(1 / 4) = 02^(a - 1) + i * 3^(1 / 4) = 02^(a - 1) + i * 3^(1 / 4) - i * 3^(1 / 4) = 0 - i * 3^(1 / 4)2^(a - 1) = -i * 3^(1 / 4)ln(2^[a - 1]) = ln(-1 * i * 3^[1 / 4])ln(2^[a - 1]) = ln(i^2 * i * 3^[1 / 4])ln(2^[a - 1]) = ln(i^3 * 3^[1 / 4])(a - 1) * ln(2) = ln(i^3 * 3^[1 / 4])(a - 1) * ln(2) / ln(2) = ln(i^3 * 3^[1 / 4]) / ln(2)(a - 1) * log_2(2) = ln(i^3) / ln(2) + ln(3^[1 / 4]) / ln(2)(a - 1) * 1 = ln(e^[i * 3 * tau / 4]) / ln(2) + (1 / 4) * ln(3) / ln(2)a - 1 = (i * 3 * tau / 4) * ln(e) / ln(2) + ln(3) / (4 * ln[2])a - 1 = (i * 3 * tau / 4) * 1 / ln(2) + ln(3) / (4 * ln[2])a - 1 = i * 3 * tau / (4 * ln[2]) + log_2(3) / 4a - 1 + 1 = i * 3 * tau / (4 * ln[2]) + log_2(3) / 4 + 1a = i * 3 * tau / (4 * ln[2]) + log_2(3) / 4 + 1a1 = i * 0 * tau / (4 * ln[2]) + log_2(3) / 4 + 1a2 = i * 1 * tau / (4 * ln[2]) + log_2(3) / 4 + 1a3 = i * 2 * tau / (4 * ln[2]) + log_2(3) / 4 + 1a4 = i * 3 * tau / (4 * ln[2]) + log_2(3) / 4 + 1
Great!!! Thanks!!!!
mesmerizing.
❤😮😊
Very good!
Thank you, Cap!
Good @MATHS TUTORIAL (BL SAHU)
❤
Method is unnecessarily complicated.
4^2a=48
Take ln both sides:
2a*ln4=ln(4*4*3)
2a*ln4=2ln4+ln3
Divide both sides by 2ln4
a=1+ln3)/2ln4
But ln4=2ln2
a=1+ln3/4ln2
Entirely agree.
Including complex roots:
4^a * 4^a = 48
4^(a + a) = (48^[1 / 2])^2
4^(2 * a) = ([2^4 * 3]^[1 / 2])^2
4^(a * 2) = ([2^4]^[1 / 2] * 3^[1 / 2])^2
(4^a)^2 = (2^[4 * (1 / 2)] * 3^[1 / 2])^2
(4^a)^2 = (2^2 * 3^[1 / 2])^2
(4^a)^2 = (4 * 3^[1 / 2])^2
sqrt([4^a]^2) = +/- sqrt([4 * 3^(1 / 2)]^2)
4^a = +/- 4 * 3^(1 / 2)
(1 / 4) * 4^a = +/- (1 / 4) * 4 * 3^(1 / 2)
4^a * 4^(-1) = +/- 4^1 * 4^(-1) * 3^(1 / 2)
4^(a - 1) = +/- 4^(1 - 1) * 3^(1 / 2)
4^(a - 1) = +/- 4^0 * 3^(1 / 2)
4^(a - 1) = +/- 1 * 3^(1 / 2)
4^(a - 1) = +/- 3^(1 / 2)
(2^2)^(a - 1) = +/- ([3^(1 / 2)]^[1 / 2])^2
2^(2 * [a - 1]) = +/- (3^[(1 / 2) * (1 / 2)])^2
2^([a - 1] * 2) = +/- (3^[1 / 4])^2
(2^[a - 1])^2 = +/- (3^[1 / 4])^2
(2^[a - 1])^2 = +(3^[1 / 4])^2, or (2^[a - 1])^2 = -(3^[1 / 4])^2
Let x = 2^(a - 1), and y = 3^(1 / 4)
(2^[a - 1])^2 = (3^[1 / 4])^2, or (2^[a - 1])^2 = -(3^[1 / 4])^2
=> x^2 = y^2, or x^2 = -y^2
=> x^2 - y^2 = y^2 - y^2, or x^2 + y^2 = -y^2 + y^2
=> x^2 - y^2 = 0, or x^2 + y^2 = 0
=> (x - y)(x + y) = 0, or (x - i * y)(x + i * y) = 0
=> (2^[a - 1] - 3^[1 / 4])(2^[a - 1] + 3^[1 / 4]) = 0, or (2^[a - 1] - i * 3^[1 / 4])(2^[a - 1] + i * 3^[1 / 4]) = 0
Suppose 2^(a - 1) - 3^(1 / 4) = 0
2^(a - 1) - 3^(1 / 4) = 0
2^(a - 1) - 3^(1 / 4) + 3^(1 / 4) = 0 + 3^(1 / 4)
2^(a - 1) = 3^(1 / 4)
ln(2^[a - 1]) = ln(3^[1 / 4])
(a - 1) * ln(2) = (1 / 4) * ln(3)
(a - 1) * ln(2) / ln(2) = ln(3) / (4 * ln[2])
(a - 1) * log_2(2) = log_2(3) / 4
(a - 1) * 1 = log_2(3) / 4
a - 1 + 1 = log_2(3) / 4 + 1
a = log_2(3) / 4 + 1
Suppose 2^(a - 1) + 3^(1 / 4) = 0
2^(a - 1) + 3^(1 / 4) - 3^(1 / 4) = 0 - 3^(1 / 4)
2^(a - 1) = -3^(1 / 4)
ln(2^[a - 1]) = ln(-1 * 3^[1 / 4])
ln(2^[a - 1]) = ln(i^2 * 3^[1 / 4])
(a - 1) * ln(2) = ln(i^2 * 3^[1 / 4])
(a - 1) * ln(2) / ln(2) = ln(i^2 * 3^[1 / 4]) / ln(2)
(a - 1) * log_2(2) = ln(i^2) / ln(2) + ln(3^[1 / 4]) / ln(2)
(a - 1) * 1 = ln(e^[i * tau / 2]) / ln(2) + log_2(3^[1 / 4])
a - 1 = (i * tau / 2) * ln(e) / ln(2) + log_2(3^[1 / 4])
a - 1 = (i * tau / 2) * 1 / ln(2) + log_2(3^[1 / 4])
a - 1 = i * tau / (2 * ln[2]) + (1 / 4) * log_2(3)
a - 1 + 1 = i * tau / (2 * ln[2]) + log_2(3) / 4 + 1
a = i * tau / (2 * ln[2]) + log_2(3) / 4 + 1
a = i * 2 * tau / (2^2 * ln[2]) + log_2(3) / 4 + 1
a = i * 2 * tau / (4 * ln[2]) + log_2(3) / 4 + 1
Suppose 2^(a - 1) - i * 3^(1 / 4) = 0
2^(a - 1) - i * 3^(1 / 4) = 0
2^(a - 1) - i * 3^(1 / 4) + i * 3^(1 / 4) = 0 + i * 3^(1 / 4)
2^(a - 1) = i * 3^(1 / 4)
ln(2^[a - 1]) = ln(i * 3^[1 / 4])
(a - 1) * ln(2) = ln(i * 3^[1 / 4])
(a - 1) * ln(2) / ln(2) = ln(i * 3^[1 / 4]) / ln(2)
(a - 1) * log_2(2) = ln(i) / ln(2) + ln(3^[1 / 4]) / ln(2)
(a - 1) * 1 = ln(e^[i * tau / 4]) / ln(2) + (1 / 4) * ln(3^[1 / 4]) / ln(2)
a - 1 = (i * tau / 4) * ln(e) / ln(2) + ln(3) / (4 * ln[2])
a - 1 = (i * tau / 4) * 1 / ln(2) + ln(3) / (4 * ln[2])
a - 1 = i * tau / (4 * ln[2]) + ln(3) / (4 * ln[2])
a - 1 = i * tau / (4 * ln[2]) + log_2(3) / 4
a - 1 + 1 = i * tau / (4 * ln[2]) + log_2(3) / 4 + 1
a = i * 1 * tau / (4 * ln[2]) + log_2(3) / 4 + 1
Suppose 2^(a - 1) + i * 3^(1 / 4) = 0
2^(a - 1) + i * 3^(1 / 4) = 0
2^(a - 1) + i * 3^(1 / 4) - i * 3^(1 / 4) = 0 - i * 3^(1 / 4)
2^(a - 1) = -i * 3^(1 / 4)
ln(2^[a - 1]) = ln(-1 * i * 3^[1 / 4])
ln(2^[a - 1]) = ln(i^2 * i * 3^[1 / 4])
ln(2^[a - 1]) = ln(i^3 * 3^[1 / 4])
(a - 1) * ln(2) = ln(i^3 * 3^[1 / 4])
(a - 1) * ln(2) / ln(2) = ln(i^3 * 3^[1 / 4]) / ln(2)
(a - 1) * log_2(2) = ln(i^3) / ln(2) + ln(3^[1 / 4]) / ln(2)
(a - 1) * 1 = ln(e^[i * 3 * tau / 4]) / ln(2) + (1 / 4) * ln(3) / ln(2)
a - 1 = (i * 3 * tau / 4) * ln(e) / ln(2) + ln(3) / (4 * ln[2])
a - 1 = (i * 3 * tau / 4) * 1 / ln(2) + ln(3) / (4 * ln[2])
a - 1 = i * 3 * tau / (4 * ln[2]) + log_2(3) / 4
a - 1 + 1 = i * 3 * tau / (4 * ln[2]) + log_2(3) / 4 + 1
a = i * 3 * tau / (4 * ln[2]) + log_2(3) / 4 + 1
a1 = i * 0 * tau / (4 * ln[2]) + log_2(3) / 4 + 1
a2 = i * 1 * tau / (4 * ln[2]) + log_2(3) / 4 + 1
a3 = i * 2 * tau / (4 * ln[2]) + log_2(3) / 4 + 1
a4 = i * 3 * tau / (4 * ln[2]) + log_2(3) / 4 + 1