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Exponential and Logarithmic Curves
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Figure 14-7 Graphs of the natural logarithmic
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function (at A) and the common logarithmic function (at B)
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Logarithm: example 2 Let s take the reciprocal of the independent variable x before performing the natural or common log, and then plot the graphs Figure 14-8 shows the results At A, we see the graph of the function
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y = ln (1/x) and at B, we see the graph of the function y = log10 (1/x)
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Graphs Involving Logarithmic Functions
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Figure 14-8 Graphs of the natural log of the
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reciprocal (at A) and the common log of the reciprocal (at B)
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These functions can also be written as y = ln (x 1) and y = log10 (x 1) Based on our knowledge of logarithms from algebra, we can rewrite these functions, respectively, as y = 1 ln x = ln x
Exponential and Logarithmic Curves
and y = 1 log10 x = log10 x When we raise x to the 1 power before taking the logarithm, we negate all the function values, compared to what they d be if we left x alone The x axis acts as a point reflector The domains and ranges of these reciprocal functions are the same as the domains and ranges of the original functions
Logarithm: example 3 We can create two interesting functions by multiplying the functions defined in the previous two paragraphs Let s do that, and see what the graphs look like We want to graph the functions
y = (ln x) [ln (x 1)] and y = (log10 x) [log10 (x 1)] Our knowledge of logarithms allows us to rewrite these functions, respectively, as y = (ln x)2 and y = (log10 x)2 The results are shown in Figs 14-9 and 14-10 The domains of both product functions span the entire set of positive reals The ranges of both functions are confined to the set of nonpositive reals (that is, the set of all reals less than or equal to 0)
Figure 14-9 Graph of the natural
log times the natural log of the reciprocal (solid black curve) The dashed gray curves are the graphs of the original functions
x 10
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Figure 14-10 Graph of the
common log times the common log of the reciprocal (solid black curve) The dashed gray curves are the graphs of the original functions
x 10
Logarithm: example 4 Finally, let s take the log functions we ve been working with and find their ratios, as follows:
y = (ln x) / [ ln (x 1)] and y = (log10 x) / [ log10 (x 1)] Our knowledge of logarithms allows us to simplify these, respectively, to y = (ln x) / ( ln x) = 1 and y = (log10 x) / ( log10 x) = 1 These functions are defined only if 0 < x < 1 or x > 1 The domains have holes at x = 1 because when we input 1 to either quotient, we end up dividing by 0 (Try it and see!) The ranges are confined to the single value 1
Exponential and Logarithmic Curves
Don t let them confuse you!
In some texts, natural (base-e) logs are denoted by writing log without a subscript, followed by the argument In other texts and in most calculators, log means the common (base-10) log To avoid confusion, you should include the base as a subscript whenever you write log followed by anything For example, write log 10 or log e instead of log all by itself, unless it s impractical to write the subscript You don t need a subscript when you write ln for the natural log If you aren t sure what the log key on a calculator does, you can do a test to find out If your calculator says that the log of 10 is equal to 1, then it s the common log If the log of 10 turns out to be an irrational number slightly larger than 23, then it s the natural log
Here s a challenge!
Draw graphs of the ratio functions we found in Logarithm: example 4 Be careful! They re a little tricky
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