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