To solve for x, we can first simplify the expression:
(3^x) * (2^(3/x)) = 24
Take the natural log of both sides to eliminate the exponents:
ln((3^x) * (2^(3/x))) = ln(24)
Using the properties of logarithms, this can be rewritten as:
ln(3^x) + ln(2^(3/x)) = ln(24)
Apply the power rule of logarithms:
x ln(3) + (3/x) ln(2) = ln(24)
Now we have a linear equation in terms of x. Rearrange the terms:
x ln(3) + 3 (ln(2) / x) = ln(24)
Multiply through by x to clear the fraction:
x^2 ln(3) + 3 ln(2) = x * ln(24)
Rearrange and factor:
x^2 ln(3) - x ln(24) + 3 * ln(2) = 0
This is a quadratic equation in terms of x, which can be solved using the quadratic formula:
x = (-(-ln(24)) ± √((-ln(24))^2 - 4 ln(3) 3 ln(2))) / (2 ln(3))
Calculating this expression will give you the values of x that satisfy the original equation.
To solve for x, we can first simplify the expression:
(3^x) * (2^(3/x)) = 24
Take the natural log of both sides to eliminate the exponents:
ln((3^x) * (2^(3/x))) = ln(24)
Using the properties of logarithms, this can be rewritten as:
ln(3^x) + ln(2^(3/x)) = ln(24)
Apply the power rule of logarithms:
x ln(3) + (3/x) ln(2) = ln(24)
Now we have a linear equation in terms of x. Rearrange the terms:
x ln(3) + 3 (ln(2) / x) = ln(24)
Multiply through by x to clear the fraction:
x^2 ln(3) + 3 ln(2) = x * ln(24)
Rearrange and factor:
x^2 ln(3) - x ln(24) + 3 * ln(2) = 0
This is a quadratic equation in terms of x, which can be solved using the quadratic formula:
x = (-(-ln(24)) ± √((-ln(24))^2 - 4 ln(3) 3 ln(2))) / (2 ln(3))
Calculating this expression will give you the values of x that satisfy the original equation.