To solve the first equation 10^x - 8.5^x = 0, we can write it as:
10^x = 8.5^x
Taking the natural log of both sides, we get:
x ln(10) = x ln(8.5)
Dividing by x on both sides (assuming x is not equal to 0), we get:
ln(10) = ln(8.5)
This equation does not have any real solutions, as ln(10) is not equal to ln(8.5).
Moving on to the second equation 16^x - 12*4^x - 64 = 0, we can factor this expression as:
(4^x - 8)(4^x - 4) = 0
This gives us two equations:
1) 4^x - 8 = 04^x = 8x = log(8)/log(4)x = 3/2
2) 4^x - 4 = 04^x = 4x = log(4)/log(4)x = 1
Therefore, the solutions to the second equation 16^x - 12*4^x - 64 = 0 are x = 3/2 and x = 1.
To solve the first equation 10^x - 8.5^x = 0, we can write it as:
10^x = 8.5^x
Taking the natural log of both sides, we get:
x ln(10) = x ln(8.5)
Dividing by x on both sides (assuming x is not equal to 0), we get:
ln(10) = ln(8.5)
This equation does not have any real solutions, as ln(10) is not equal to ln(8.5).
Moving on to the second equation 16^x - 12*4^x - 64 = 0, we can factor this expression as:
(4^x - 8)(4^x - 4) = 0
This gives us two equations:
1) 4^x - 8 = 0
4^x = 8
x = log(8)/log(4)
x = 3/2
2) 4^x - 4 = 0
4^x = 4
x = log(4)/log(4)
x = 1
Therefore, the solutions to the second equation 16^x - 12*4^x - 64 = 0 are x = 3/2 and x = 1.