Solve:
log2(2x^2)[log2(16x)]= 9/2log2x
To solve the given equation:
1. Let's simplify the equation step by step.
2. Start by simplifying the left side of the equation using the properties of logarithms:
log2(2x^2)[log2(16x)]= log2(2x^2) * log2(16x)
3. Apply the logarithmic properties to further simplify:
log2(2x^2) = 2 log2(x)
log2(16x) = log2(16) + log2(x)
4. Use the property logb(a^n) = n*logb(a) to simplify further:
2 log2(x) * (log2(16) + log2(x))
Log2(16) = 4, as 2^4 = 16
2 log2(x) * (4 + log2(x))
5. Multiply 2 by 4 and then distribute 2 log2(x) to both terms inside the parentheses:
8 log2(x) + 2(log2(x))^2
6. Now, let's simplify the right side of the equation:
9/2 log2(x) = (9/2) * log2(x)
7. Setting the left side equal to the right side:
8 log2(x) + 2(log2(x))^2 = (9/2) * log2(x)
8. Next, let's solve this quadratic equation by bringing all the terms to one side:
2(log2(x))^2 - (9/2)*log2(x) + 8*log2(x) = 0
2(log2(x))^2 + (8 - (9/2))log2(x) = 0
2(log2(x))^2 + (16/2 - 9/2)log2(x) = 0
2(log2(x))^2 + (7/2)log2(x) = 0
9. Set a new variable, let's say u = log2(x):
2u^2 + (7/2)u = 0
10. Factor out u:
u(2u + 7/2) = 0
11. Set each factor equal to zero:
u = 0 OR 2u + 7/2 = 0
12. Solve for u in the second equation:
2u + 7/2 = 0
2u = -7/2
u = -7/4
13. Substitute the value of u back in terms of x:
log2(x) = 0 OR log2(x) = -7/4
14. Solve for x:
For log2(x) = 0:
x = 2^0
x = 1
For log2(x) = -7/4:
x = 2^(-7/4)
x = 1/2^(7/4)
x = 1/2^(7/4)
x = 1/2^(7/4)
x ≈ 0.231
15. The two solutions to the equation are x = 1 and x ≈ 0.231.