You first must translate the logarithm to exponential form:
y=log2(x-4) ——> 2^y=x-4
Now, we have an equivalent exponential function. To find the y-intercept, input x=0 into the function since we want to find the output when x has no movement.
2^y=((0)-4)
2^y=-4
Therefore, there is no y-intercept because we can’t input a negative value into the function and 0 is the lowest value. This is because the function is translated 4 units right, hence the “x-4” in the logarithm. Remember, translations always have opposite signs, so - means right and + means left. Now, we must look at the parent function:
y=log2(x) is the parent function, or:
2^y=x
If we want to find the y-intercept of the parent function, we input 0 for x:
2^y=0
There is a problem here. No value for y will ever result in 0. There is no exponent that equals 0. So, the parent function is used in the translated function. This means that this rule applies, except the function’s domain is shifted 4 units right. However, you can use the parent function to answer your question since the translated function originates from it.
Put simply, this function doesn’t have a y-intercept because there is no exponent that will produce an output of 0, which creates an asymptote of x=0.