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The twin of exponential equations are logarithms. |
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| If we say ... | ||
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Y = eX |
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| That also means ... | ||
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log eY = X |
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(also known as ln Y = X) |
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| Remember the way you read a log is: | ||
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| Which is the same thing we said in the beginning ... | ||
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Y = eX |
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(NOTE: From here on we'll be using ln instead of loge) |
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| So the point is: | ||
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Y = eX and ln Y = X |
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| Say the exact same thing. | ||
| So let's talk about the derivatives and integrals of things like | ||
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ln X |
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| When we went looking for the derivative of eX, we found that it was eX, | ||
| give or take a dX. | ||
| Wouldn't it be nice if ln X worked the same way | ||
| SORRY, No such luck. | ||
| But finding it isn't too bad. | ||
| Start with | ||
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Y = ln X |
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| which is the same thing as saying: | ||
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X = eY |
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(Don't worry, we'll be back to ln X before we're done) |
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| OK, the derivative of eY (with respect to the variable Y) is eY | ||
| so we can say: | ||
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| We can flip these over and say: | ||
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| Now we do some substitution. | ||
| We started with the equations: | ||
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Y = ln X also known as X = eY |
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| Now we substitute both of these into our last equation: | ||
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| In English, what this says is that the derivative of lnX, | ||
| is equal to 1/X (also known as X-1) | ||
| This also means that the integral of X-1 is lnX, | ||
| remembering, of course, to include dX's and + C's where needed. | ||
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Sometimes
you will see  written as  |
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| Don't let that throw you, they both mean the same thing. | ||
| Examples: | ||
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| Did you notice the 2X in the numerator of the last one? | ||
| The derivative of X2 is 2XdX, so everything we needed to have, | ||
| to "Put Back Into" the 1/X2 was there. | ||
| So this all works out great for things like eX and ln X, | ||
| but what if you have to deal with 2X or log8X ? | ||
| Huh??? | ||
| Well????? | ||
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copyright 2005 Bruce Kirkpatrick |
