4^(cos^2(x)) + 4^(cos^2(x)) = 3
Simplify:
2(4^(cos^2(x))) = 3
Divide by 2:
4^(cos^2(x)) = 1.5
Taking the natural logarithm of both sides:
ln(4^(cos^2(x))) = ln(1.5)
Using the property of logarithms:
cos^2(x) * ln(4) = ln(1.5)
Solving for cos^2(x):
cos^2(x) = ln(1.5) / ln(4)
cos^2(x) ≈ 0.7023
Now, take the square root of both sides to find cos(x):
cos(x) ≈ ±√0.7023
cos(x) ≈ ±0.8384
Therefore, the solutions for x are:
x ≈ arccos(0.8384) or x ≈ arccos(-0.8384)
4^(cos^2(x)) + 4^(cos^2(x)) = 3
Simplify:
2(4^(cos^2(x))) = 3
Divide by 2:
4^(cos^2(x)) = 1.5
Taking the natural logarithm of both sides:
ln(4^(cos^2(x))) = ln(1.5)
Using the property of logarithms:
cos^2(x) * ln(4) = ln(1.5)
Solving for cos^2(x):
cos^2(x) = ln(1.5) / ln(4)
cos^2(x) ≈ 0.7023
Now, take the square root of both sides to find cos(x):
cos(x) ≈ ±√0.7023
cos(x) ≈ ±0.8384
Therefore, the solutions for x are:
x ≈ arccos(0.8384) or x ≈ arccos(-0.8384)