9^log3logx = log x -2log^2x + 4
Assuming you meant:
(9^log3)logx = log x - 2log^2 x + 4
let logx = k , then you have
(9^log3)k = k - 2k^2 + 4
2k^2 + k(9^log3 - 1) - 4 = 0
looks like some calculator work ahead
use the quadratic formula where
a = 2 , b = 9^log3, and c = -4
to find k
remember k = logx , then find x
Are all the logs to base 10?
Do you mean 9^(log3*logx) or 9^log3 * logx ?
And does log^2x mean (logx)^2 ?
One way of reading your equation is
9^log3 logx = logx - 2(logx)^2 + 4
which is just a quadratic in logx:
2(logx)^2 + (9^log3 - 1) logx - 4 = 0
so, if you let u=logx, then that is just
2u^2 + 1.853u - 4 = 0
which is easy to solve.
Somehow I think you have mangled the equation.
To solve this equation, we'll start by applying some logarithmic properties to simplify the equation.
First, we'll rewrite the equation in terms of logarithms with the same base. Let's assume the base is 10:
9^(log3logx) = log(x) - 2log^2(x) + 4
Next, we'll use the property that states log_a(b^c) = c * log_a(b). Applying this property to the left side of the equation, we have:
= log(3logx * log(9))
Now, let's distribute the logarithm on the right side:
= log(x) - 2log(x) + 8
Combining like terms, we simplify further:
= log(x) - log(x^2) + 8
Now, we can combine the logarithms on the right side using the quotient rule of logarithms. The quotient rule states that log_a(b/c) = log_a(b) - log_a(c):
= log(x/x^2) + 8
= log(1/x) + 8
Next, we'll combine the logarithms using the sum rule, which states that log_a(b) + log_a(c) = log_a(b * c):
= log([1/x] * 10^8)
= log(10^8 / x)
= log(10^8) - log(x)
= 8 - log(x)
Now, the simplified equation is:
log(3logx * log(9)) = 8 - log(x)
To solve for x, we'll need to use properties of logarithms to eliminate the logs. Let's solve step by step:
1. Move the logs to one side:
log(3logx * log(9)) + log(x) = 8
2. Combine the logs on the left side using the product rule, which states that log_a(b) + log_a(c) = log_a(b * c):
log[(3logx * log(9)) * x] = 8
log(3x * logx * log(9)) = 8
3. Apply the exponentiation rule of logarithms, which states that log_a(b) = c is equivalent to a^c = b:
3x * log(x) * log(9) = 10^8
4. Simplify further by dividing both sides by (3 * log(x) * log(9)):
x = 10^8 / (3 * log(x) * log(9))
This is the simplified solution for x.