2^(2x) - 14*2^x - 32 = 0 Let's denote 2^x as a. Then we get: a^2 - 14a - 32 = 0 (a - 16)(a + 2) = 0 a = 16 or a = -2 2^x = 16 or 2^x = -2 x = log_2(16) or x = log_2(-2) x = 4 or x is not a real number
5^(2x) - 3 + 5^x - 1 = 250 Let's denote 5^x as a. Then we get: a^2 - 3 + a - 1 = 250 a^2 + a - 4 = 250 a^2 + a - 254 = 0 (a - 13)(a + 17) = 0 a = 13 or a = -17 5^x = 13 or 5^x = -17 x = log_5(13) or x is not a real number
Therefore, the solutions to the exponential equations are x = log_5(14) / 6, x = 4, and x = log_5(13).
25^(3x) - 10 = 4
25^(3x) = 14
(5^2)^(3x) = 14
5^(2*3x) = 14
5^(6x) = 14
6x = log_5(14)
x = log_5(14) / 6
2^(2x) - 14*2^x - 32 = 0
Let's denote 2^x as a. Then we get:
a^2 - 14a - 32 = 0
(a - 16)(a + 2) = 0
a = 16 or a = -2
2^x = 16 or 2^x = -2
x = log_2(16) or x = log_2(-2)
x = 4 or x is not a real number
5^(2x) - 3 + 5^x - 1 = 250
Let's denote 5^x as a. Then we get:
a^2 - 3 + a - 1 = 250
a^2 + a - 4 = 250
a^2 + a - 254 = 0
(a - 13)(a + 17) = 0
a = 13 or a = -17
5^x = 13 or 5^x = -17
x = log_5(13) or x is not a real number
Therefore, the solutions to the exponential equations are x = log_5(14) / 6, x = 4, and x = log_5(13).