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# 4.4.notes - 10 y as the value of the exponent when y is...

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MAC 2233/001-006 Business Calculus, Fall 2011 CRN: 81700–81702, 81705, 81716, 81715 Assistants : Matthew Fleeman, Arbin Rai, Tadesse Zerihun, Vindya Kumari, Junyi Tu, Jing- han Meng On exponential and logarithmic functions All bases in this note are positive. I shall give a heuristical reasoning (understanding) of logarithmic and exponential functions. Consider base 2 as an example. The function y = 2 x is an exponential function because the “variable” x is in the exponent. Note that the functions x 2 and 2 x behave very differently because 2 x grows very fast (as x → ∞ ) and we call it exponential growth. This phenomenon is captured in the following diagram where the blue graph is for y = 2 x and the red one is for y = x 2 . The logarithmic functions are inverse functions of exponential functions (with the same base). So y = 10 x if and only if x = log 10 ( y ) . We should therefore interpret log

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Unformatted text preview: 10 ( y ) as the value of the exponent when y is expressed in base 10 . Such interpretation gives that 10 log 10 ( x ) = x, and log 10 (10 x ) = x. In addition, with the laws of exponents, we see that log 10 ( xy ) = exponent when xy is written in base 10 = sum of exponents when x and y are written in base 10 = log 10 ( x ) + log 10 ( y ) . Similarly, we have log 10 ( x/y ) = log 10 ( x ) − log 10 ( y ) , and log 10 ( x r ) = r log 10 ( x ) . Therefore, log 10 ( x + y ) n = log 10 ( x ) + log 10 ( y ) , and log 10 ( x − y ) n = log 10 ( x ) − log 10 ( y ) . The above discussion also applies to other bases such as 2, e. I give below a diagram of the graphs of y = exp( x ) and its inverse y = ln( x ). y x 4 4 2 2-2-4-2-4 y = exp(x) y = ln(x) y=x...
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