Video Transcript
Determine the point at which the graph of the function 𝑓 of 𝑥 equals log to the base six of 18 minus 𝑥 intersects the 𝑥-axis.
So what we can see first of all is that we’ve got a function 𝑓 of 𝑥 that’s equal to log to the base six of 18 minus 𝑥. Well, what we can do because we’re looking at logarithms is look at a couple of relationships that we know. So if we have 𝑐 is equal to log to the base 𝑏 of 𝑎, so we have our logarithm in this form, then what we can say is that 𝑎 is gonna be equal to 𝑏 to the power of 𝑐. And we’re gonna be able to use this to help us solve our problem. But what other information are we given in the question?
Well, what we’re also looking at is the point at which the graph of the function intersects the 𝑥-axis. Well, what we know about this is that the 𝑥-value we don’t know. But we do know the 𝑦-value because this is gonna be zero. Because if we’re intersecting the 𝑥-axis, then it’s gonna be zero.
So now what we know is that our 𝑎 is gonna be 18 minus 𝑥, our 𝑏 is gonna be six, and our 𝑐 is gonna be zero because that’s the value of the function at the point of 𝑥 we’re looking for. So therefore, we can say that zero is equal to log to the base six of 18 minus 𝑥. So now if we apply the rule that we looked at at the beginning, then we can say that 18 minus 𝑥 is equal to six to the power of zero.
Well, we can also have a look here at one of our exponent rules. And that is that anything to the power of zero is equal to one. So, for example, 𝑥 to the power of zero is equal to one. So therefore, we know that six to the power of zero is also gonna be equal to one. So therefore, what we have is that 18 minus 𝑥 is equal to one. So now if we add 𝑥 and subtract one from each side of the equation, what we’re gonna have is 17 is equal to 𝑥. So have we solved the problem?
Well, not quite, cause we’ve just found the value of 𝑥. What we’re trying to do is determine the point at which the graph of the function 𝑓 of 𝑥 equals log to the base six of 18 minus 𝑥 intersects the 𝑥-axis. So what this is going to be is the point 17, zero.
