Find the domain of definition of function f(x) given by
f(x)= log base4{log base5(log base3(18x - x^2 - 77)}
Let u=18x-x^2-77
log_3(u) exists if u>0
log_5(log_3)u)) exists if log_3(u)>0, or u > 1
log_4(log_5(log_3(u))) exists if log_5(log_3(u)) > 0, or log_3(u) > 1, or u > 3
So, we need
18x-x^2-77 > 3
18x-x^2-80 > 0
(10-x)(x-8) > 0
8 < x < 10
To find the domain of definition of the function f(x) = log base4{log base5(log base3(18x - x^2 - 77))}, we need to consider the restrictions imposed by the logarithmic functions.
First, let's start from the innermost logarithmic function, which is log base3(18x - x^2 - 77). The logarithm is defined only for positive arguments, so we need to ensure that the expression inside the logarithm is greater than zero:
18x - x^2 - 77 > 0
To find the values of x that satisfy this inequality, we can solve it by finding the roots:
18x - x^2 - 77 = 0
Rearranging the equation, we have:
x^2 - 18x + 77 = 0
Using the quadratic formula, we can find the roots of this equation:
x = (18 ± √(18^2 - 4*1*77)) / 2
x = (18 ± √(324 - 308)) / 2
x = (18 ± √16) / 2
x = (18 ± 4) / 2
x1 = (18 + 4) / 2 = 22 / 2 = 11
x2 = (18 - 4) / 2 = 14 / 2 = 7
So the roots of the equation are x = 11 and x = 7.
Now, let's consider the next logarithmic function: log base5(log base3(18x - x^2 - 77)). To ensure that this logarithm is defined, we need the argument to be positive:
log base3(18x - x^2 - 77) > 0
Since we have already found the values of x that make the expression inside the logarithm positive, we can use these intervals to determine the domain. In this case, the domain is defined for x < 7 and x > 11.
Finally, we consider the outermost logarithmic function: log base4{log base5(log base3(18x - x^2 - 77))}. For this logarithm to be defined, its argument must be positive:
log base5(log base3(18x - x^2 - 77)) > 0
Since the argument must be positive, we can exclude x = 7 and x = 11 from the domain.
Therefore, the domain of definition for the function f(x) = log base4{log base5(log base3(18x - x^2 - 77))} is x < 7 or x > 11.