To solve this equation:
9^2 + 1 = 81^(x - 2) / (3x)
First, simplify the equation:
81 + 1 = (3^(2x - 2)) / (3x)
Now, combine like terms:
82 = 3^(2x - 2) / (3x)
To solve for x, we'll use logarithms. Take the natural logarithm (ln) of both sides:
ln(82) = ln(3^(2x - 2) / (3x))
Using the properties of logarithms, we can rewrite this as:
ln(82) = ln(3^(2x - 2)) - ln(3x)
Now, apply the power rule of logarithms, which allows you to bring the exponent down as a coefficient:
ln(82) = (2x - 2) * ln(3) - ln(3x)
Now, we'll isolate the terms involving x on one side of the equation:
ln(82) + ln(3x) = (2x - 2) * ln(3)
Now, distribute ln(3) to both terms on the right side:
ln(82) + ln(3x) = 2x * ln(3) - 2 * ln(3)
Now, add 2ln(3) to both sides:
ln(82) + ln(3x) + 2ln(3) = 2x * ln(3)
Now, divide by 2ln(3):
(x * ln(3)) = (ln(82) + ln(3x) + 2ln(3)) / 2ln(3)
Finally, divide by ln(3) to isolate x:
x = (ln(82) + ln(3x) + 2ln(3)) / (2ln(3))
The solution for x is quite complex, and it involves logarithmic terms. You may want to use a calculator or a computer program to approximate the value of x.