Express y as a function of x. What is the domain? Explain your answer.
log4y = x + log4
To express y as a function of x, we need to isolate y on one side of the equation.
Initially, we have the equation: log4y = x + log4
To isolate y, we can start by subtracting log4 from both sides:
log4y - log4 = x
Using logarithmic properties, we can simplify the equation using the rule log(a) - log(b) = log(a/b):
log4(y/4) = x
Now, we can exponentiate both sides with base 4 to eliminate the logarithm:
4^(log4(y/4)) = 4^x
By definition, when a logarithm and its base are the same, they cancel each other out. Therefore, we are left with:
y/4 = 4^x
Now, let's solve for y:
y = 4 * 4^x
So, the expression for y as a function of x is y = 4 * 4^x.
Next, let's determine the domain of this function.
In this case, since we do not have any restrictions on the value of x or y, the domain is the set of all real numbers (-∞, ∞).
In simpler terms, any real value of x can be substituted into the function to yield a real value of y. Thus, the domain of the function is all real numbers.