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We can use a calculator to determine the values of y for various values of x Figure 14-6 is the resulting graph for values of x ranging from 10 to 10 When we input x = 0, we get e1/0, which is undefined For any other real value of x, the output value y is a positive real number Therefore, the domain of this function is the set of all nonzero reals No matter how large we want y to be when y > 1, we can always find some value of x that will give it to us Similarly, no matter how small we want y to be when 0 < y < 1, we can always find some value of x that will do the job However, we can t find any value for x that will give us y = 1 For that to happen, we must raise e to the 0th power, meaning that we must find some x such that 1/x = 0 That s impossible! Therefore, the range of our function is the set of all positive reals except 1 The graph has a horizontal asymptote whose equation is y = 1, and a vertical asymptote corresponding to the y axis The open circle at the point (0,0) indicates that it s not part of the graph
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Graph of the function y = e(1/x) Note the hole in the domain at x = 0 and the hole in the range at y = 1
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Graphs Involving Logarithmic Functions
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A logarithm (sometimes called a log) of a quantity is a power to which a positive real constant is raised to get that quantity As with exponential functions, the constant is called the base, and it s almost always equal to either e or 10 The base-e log function, also called the natural logarithm, is usually symbolized by writing ln or log e followed by the argument (the quantity on which the function operates) The base-10 log function, also called the common logarithm, is usually symbolized by writing log10 or log followed by the argument
They re inverses! A logarithmic function is the inverse of the exponential function having the same base The natural logarithmic function undoes the work of the natural exponential function and vice versa, as long as we restrict the domains and ranges so that both functions are bijections The common logarithmic and exponential functions also behave this way, so we can say that
ln ex = x = e(ln x) and log 10x = x = 10(log x) For these formulas to work, we must restrict x to positive real-number values, because the logarithms of quantities less than or equal to 0 are not defined
Logarithm: example 1 Figure 14-7 illustrates graphs of the two basic logarithmic functions operating on a variable x At A, we see the graph of the base-e logarithmic function, over the portion of the domain from 0 to 10 The equation is
y = ln x At B in Fig 14-7, we see the graph of the base-10 logarithmic function, over the portion of the domain from 0 to 10 The equation is y = log10 x As with the exponential graphs, these curves have similar contours, and they look almost identical if we choose the axis scales as we ve done here The domains of the natural and common log functions both span the entire set of positive reals When we try to take a logarithm of 0 or a negative number, however, we get a meaningless quantity (or, at least, something outside the set of reals!) By inputting just the right positive real value to a log function, we can get any real-number output we want The ranges of the log functions therefore include all real numbers
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