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anonymous

  • 5 years ago

what are logarithmic functions please try to explain

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  1. shadowfiend
    • 5 years ago
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    What do you mean exactly by `what are' they?

  2. anonymous
    • 5 years ago
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    It’s too hard to explain what logarithmic functions are in one sitting. It took me a while to understand them. And you can’t fully appreciate them until you learn what they really are at a deep level. But a logarithmic function, you can think of it as the inverse of the exponential function. For example, consider f(x) = e^x. That’s the exponential function. You can plug in values for x, and you get back values f(x). You can plug in x = 0, and you will get back f(0) = e^0 = 1. So (0, 1) is a point on f(x) = e^x. Plug some more in. Plug x = 1, and you get back f(1) = e^1 = e. (1, e) is a point on f(x). Plug x = 100, and you get f(100) = e^100. (100, e^100) is another point. Plug in x = -100, and you get back e^(-100). (-100, e^(-100)) is another point. If you try plug in all possible numbers into x for f(x) = e^x, you’ll see that you trace out a smooth, continuous curve that “grows exponentially.” The logarithmic functions are inverses of exponential functions. Let me call g(x) to be g(x) = log x, where the log has “base e,” the natural logarithm. Remember that (1, e) is a point on f(x)? Then (e, 1) is a point on g(x). That’s because g(e) = log (e) = 1. Similarly, since (100, e^100) is a point on f(x), we have (e^100, 100) is a point on g(x). That’s because g(e^100) = log (e^100) = 100 * log (e) = 100. Also, since (-100, e^(-100)) is a point on f(x), we have (e^(-100), -100) is a point on g(x). That’s because g(e^(-100)) = log (e^(-100)) = -100 * log (e) = -100. So if a point (a, b) is on f(x) = e^x, then the point (b, a) is on g(x) = log (x). So we see that they’re mirror images of each other. In fact, they’re mirror images across the line y = x since the x and y coordinates swapped. So e^x and log x are inverse functions not only because they “undo” each other, but also because they’re mirror images of each other across the line y = x. This helps us visualize log x, Graphically, inverse functions are mirror images of each other, reflected across the line y = x. So imagine drawing f(x) = e^x on a graph paper. Then draw in the line y = x. Fold your graph paper along the line y = x, and let f(x) = e^x be reflected across y = x. The reflection of e^x across the line y = x is just where the the curve e^x touches the graph paper when you folded it. That reflection is g(x) = log x.

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