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18.01 Single Variable Calculus, Fall 2007
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18.01 Single Variable Calculus, Fall 2007
Transcript – Lecture 6
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PROFESSOR: All right, so let's begin Lecture Six. We're talking today about
exponentials and logarithms. And these are the last functions that I need to
introduce, the last standard functions that we need to connect with Calculus, that
you've learned about. And they're certainly as fundamental, if not more so, than
trigonometric functions.
So first of all, we'll start out with a number, a, which is positive, which is usually
called a base. And then we have these properties that a to the power 0 is always 1.
That's how we get started. And a^1 is a. And of course a^2 , not surprisingly, is a
times a, etc. And the general rule is that a^(X1 X2) is a^X1 times a^X2 . So this is
the basic rule of exponents, and with these two initial properties, that defines the
exponential function. And then there's an additional property, which is deduced from
these, which is the composition of exponential functions, which is that you take a to
the X1 power, to the X2 power. Then that turns out to be a to the X1 times X2. So
that's an additional property that we'll take for granted, which you learned in high
school.
Now, in order to understand what all the values of a^x are, we need to first
remember that if you're taking a rational power that it's the ratio of two integers
power of a. That's going to be a ^ m, and then we're want to have to take the nth
root of that. So that's the definition. And then, when you're defining a ^ x, so a^x is
defined for all x by filling in. So I'm gonna use that expression in quotation marks,
"filling in" by continuity. This is really what your calculator does when it gives you a
to the power x, because you can't even punch in the square root of x. It doesn't
really exist on your calculator. There's some decimal expansion. So it takes the
decimal expansion to a certain length and spits out a number which is pretty close to
the correct answer. But indeed, in theory, there is an a to the power, square root of
2, even though the square root of 2 is irrational. And there's a to the pi and so forth.
All right, so that's the exponential function, and let's draw a picture of one. So we'll
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 Fall '08
 BRUBAKER
 Calculus, Chain Rule, Derivative, Thing, Natural logarithm, Logarithm

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