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Math Help - exponential and logarithmic functions

  1. #1
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    exponential and logarithmic functions

    I am working on converting logarithmic equations in exponential form but I do not understand how my book explains it.

    The two problems I am having trouble with are:

    ln e^3 = 3

    ln 1/e^2 = -2

    Could someone please explain how to do this so I understand the concept?
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  2. #2
    Super Member Matt Westwood's Avatar
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    Suppose you have an exponential equation like:
    a^b = c

    The "logarithm" function says: "What number do I have to raise a to so as to get c?"

    So we have \log_a c = b.

    Now when the number being raised to the power is e (which is of course 2.71828 ...), the same thing applies.

    So e^x = y is another way of saying \log_e y = x. And a special symbol for \log_e is \ln.

    Let's look at \ln e^3 = x, where we're trying to find x.

    This is saying: "What number do I raise e to so as to get e^3?"

    What number do I raise n to (where n is any number) to get n^3?

    As you can see, \ln{} "undoes" the work that " e to the power of" does.

    That is, logarithm and exponential (if it's the same base) are inverse functions.

    Now the second one is trickier.

    \ln \left({\frac 1 {e^2}}\right) = x

    The thing here is to remember that \ln \left({\frac 1 {x}}\right) = - \ln x (it just is, the book probably shows why).

    So \ln \left({\frac 1 {e^2}}\right) = -\ln e^2 = -2 (from what we did before).
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  3. #3
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    Thanks

    Wow!

    Thank you very much, that explanation was so much better than my book.

    I appreciate it.
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