Stony Brook MAT 123 Fall 2015
What is a logarithm?

Start   We are going to learn now, about what a logarithm is. So, we know that 10 to the 1 is 10, and 10 to the 2 is 100. So there must be a power of 10 that gives us 50. We call that power the log of 50. But we have a problem! Well, first what do we know about x? we know that
0:31x is somewhere between 1 and 2. How do we know it is between 1 and 2? Because 10 to the 1 gives us 10 and 10 to the 2 gives us 100. But we also know that for example 2^5 is 32 and 2^6 is 64. So, there must be a power of 2 that gives us a 50. And we call that the log of 50. The problem is that now x instead of being between 1 and 2, x must be between
1:065 and 6. That cannot be. They cannot both be the log of 50. So how do we tell them apart?
Well, for here we would say x is the log base 10 of 50. So we put a little 10 down there to remind us that we are taking 10 and raising it to this number and getting 50, and we know that this number is somewhere between 1 and 2. Here we should say log base 2 of 50 because
1:39now we know we are taking 2 and raising to x and getting 50 and that number is somewhere between 5 and 6 So, in general if we say the log base "b" of 50 equals x. That means if we take whatever the base is
2:06and we raise to the x, we get 50. And there is going to be of course lots of bases, so lots of different values for b. And more general, if we say the log base b of x equal a, that means if we take b and we raise it to the a we get x. Let's do a couple of examples.
2:35The log base 3 of 9 equals x means if we take 3^x you get 9. So, x must be 2. If we had the log base 4 of 64 equals x, then it is easy: if we take 4 and raise it to the x we get 64.
3:01So x must be 3 because 4^3 is 64. So one thing we figure out about logs, we can get nice simple numbers, we also realize we can get not so simple numbers, because the log of 50 is a number log base 10 of 50 is a number between 1 and 2, but it is not immediately obvious what that number is.
What if we wanted to do the log base 4 of 1/16?
3:35That means if we take 4 and raise it to x, we get 1/16. Well, 4 to the 2 is 16. So 4 to the negative 2 would give us 1/16, because 1/16 is 4 to the negative 2.
So logs can come out negative as well as coming out positive. Can we get zero? Sure!
4:03What about the log base b of 1, that equals zero because any number to the zero equals 1.
Except maybe for zero. So, logs can come out positive numbers, they can come out negative numbers, they can come out zero. So a log can come out to be anything we want. However, we cannot take the log of a negative number. Because imagine we are doing the log base b of -5 equals x. We would say:
4:33what can we raise b to to get -5? And the answer is: there is no value. As long as b is a positive number there is nothing we can it raise to so we get -5. So we know that for logs the domain is going to be x has to be greater than zero, the range is going to be all reals. Since a couple of other important things to take away form logs. First, the
5:05log base b of x equals a means b^a equals x. The second thing is the log base b of 1 equals zero, because b raised to the zero always gives you 1. Finally, the log base b of b equals 1, because if i take b and raise it to 1, I will get b. I said finally, there
5:34is one more. The log base b of b^x equals x, because if I take b I raise it to the x to get b^x. And this is going to help us realize that the log and the exponential are inverses of each other, but that is for another module.