logx(9x^2)[log3(x)]^2=4
i got the x=1/9 and x=3 is that correct?
Hard to say. Your syntax is murky. How did you arrive at your answer?
You appear to have log(base x) of 9x^2 or log_10(x) * 9x^2
and log(base 3) of x
And what happens when you multiply logs?
When you divide logs, you change base...
If the question is
logx(9x^2)[log3 (x)]^2=4
then x = 3 , 1/9 satisfies the equation.
As oobleck asked, how did you get that, other than by inspection.
I went as far as
(2 log3/logx + 2)(logx/log3)^2 = 4
when x = 3, the first bracket becomes (4) and the second bracket becomes 1, so it works
if x = 1/9, the first bracket becomes (1) and the 2nd becomes (-2)^2, so it works
Nice work, Reiny. Using u = log3/logx, we could have proceeded with
(2u+2)/u^2 = 4
u+1 = 2u^2
2u^2-u-1 = 0
(2u+1)(u-1) = 0
u = -1/2 or 1
So, that means that log3/logx = -1/2 or 1
logx/log3 = -2 or 1
logkx = -2 or 1
x = 3^-2 or 3^1
x = 1/9 or 3
To solve the equation logx(9x^2)[log3(x)]^2 = 4, we need to carefully follow the given steps:
Step 1: Start by simplifying the equation:
Rewrite [log3(x)]^2 as log3(x) * log3(x), and then substitute log3(x) with a new variable, let's say y.
So the equation becomes logx(9x^2) * y^2 = 4.
Step 2: Use the property of logarithms to rewrite the equation:
logx(9x^2) can be rewritten as logx(9) + logx(x^2).
Substituting y back in for log3(x), we can also replace logx(x^2) with 2logx(x) = 2.
Therefore, the equation becomes logx(9) + 2y^2 = 4.
Step 3: Express everything in terms of a single logarithm:
Use the change of base formula to convert logx(9) to log3(9)/log3(x), and then use the substitution y = log3(x) to rewrite it as 2y^2 + log3(9)/y^2 = 4.
Step 4: Simplify the equation further:
Multiply through by y^2 to get rid of the denominators, resulting in 2y^4 + log3(9) = 4y^2.
Step 5: Rearrange the equation:
Subtract 4y^2 from both sides, resulting in 2y^4 - 4y^2 + log3(9) = 0.
Step 6: Solve for y:
To solve this quadratic-like equation in terms of y^2, let z = y^2, giving us 2z^2 - 4z + log3(9) = 0.
Solve this quadratic equation for z, and obtain two possible values for z, let's say z1 and z2.
Step 7: Find the corresponding values for y and x:
Using z1 and z2, calculate y1 = √z1 and y2 = √z2.
Substitute each value of y back into y = log3(x) to find two possible values for x, let's say x1 and x2.
After following these steps, you will find the correct values for x that satisfy the equation logx(9x^2)[log3(x)]^2 = 4.