Which is the domain of the function y = log 3 ( 2^x - 16 ) ?
y= log 3 (2^x - 16)
We know that the domain of y are all x values where y is defined.
The lorarithm is defined if 2^x - 16 > 0
==> 2^x - 16 >0
==> 2^x > 16
==> 2^x > 2^4
==> x > 4
Then the domain is when x belongs to the interval (4, inf)
The domain of the given function is restrictive. We'll determine the domain of the function by imposing the constraints of existance of logarithm function.
2^x - 16 > 0
We'll isolate x to the left side.
We'll add 16 both sides and we'll get:
2^x > 16
We'll write 16 as a power of 2 and we'll have:
16 = 2^4
We'll re-write the inequality:
2^x > 2^4
Because the bases ar matching, we can apply the one to one property of logarithms. Because the base is >1, the direction of the inequality remains unchanged.
x > 4
The domain of definition of the function y = log 3 ( 2^x - 16 ) is the interval (4 , +infinite).
y = log3 (2^x-16).
To find the domain of x.
The image set y is real as long as log3 (2^x -16) is real.
Therefore the antilog y = 2^x-16 should be positive.
2^x -16 > 16
x log2 > 16
x > log16/log2 = 4
So x > 4.
Or the domain of x is ] 4, infinity [, which means x > 4. x not equal to 4.