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Figure 14-11 is a graph of the function y = (ln x) / [ln (x 1)]
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natural log divided by the natural log of the reciprocal (solid black line with holes ) The dashed gray curves are the graphs of the original functions
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Points (0, 1) and (1, 1) are not part of the graph of the ratio function!
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common log divided by the common log of the reciprocal (solid black line with holes ) The dashed gray curves are the graphs of the original functions
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1 Points (0, 1) and (1, 1) are not part of the graph of the ratio function!
Figure 14-12 is a graph of the function y = (log10 x) / [log10 (x 1)] In both graphs, the original numerator and denominator functions are graphed as dashed gray curves The ratio functions are graphed as solid black lines with holes The small open circles at the points (0, 1) and (1, 1) indicate that those points are not part of either graph That s the trick I warned you about Without the open circles, these graphs would be wrong
Logarithmic Coordinate Planes
Engineers and scientists sometimes use coordinate systems in which one or both axes are graduated according to the common (base-10) logarithm of the displacement Let s look at the three most common variants
Semilog (x -linear) coordinates Figure 14-13 shows semilogarithmic (semilog) coordinates in which the independent-variable axis is linear, and the dependent-variable axis is logarithmic The values that can be depicted on the y axis are restricted to one sign or the other (positive or negative) The graphable intervals in this example are
1 x 1 and 01 y 10
Exponential and Logarithmic Curves
Figure 14-13 The semilog
coordinate plane with a linear x axis and a logarithmic y axis
x 1 0 1
The y axis in Fig 14-13 spans two orders of magnitude (powers of 10) The span could be increased to encompass more powers of 10, but the y values can never extend all the way down to 0
Semilog (y -linear) coordinates Figure 14-14 shows semilog coordinates in which the independent-variable axis is logarithmic, and the dependent-variable axis is linear The values that can be depicted on the x axis are restricted to one sign or the other (positive or negative) The graphable intervals in this illustration are
01 x 10 and 1 y 1 The x axis in Fig 14-14 spans two orders of magnitude The span could cover more powers of 10, but in any case the x values can t extend all the way down to 0
Log-log coordinates Figure 14-15 shows log-log coordinates Both axes are logarithmic The values that can be depicted on either axis are restricted to one sign or the other (positive or negative) In this example, the graphable intervals are
01 x 10 and 01 y 10 Both axes in Fig 14-15 span two orders of magnitude The span of either axis could cover more powers of 10, but neither axis can be made to show values down to 0
Logarithmic Coordinate Planes
Figure 14-14 The semilog
coordinate plane with a logarithmic x axis and a linear y axis
0 03 1 3 10
Figure 14-15 The log-log
coordinate plane The x and y axes are both logarithmic
01 01
x 03 1 3 10
Are you confused
Semilog and log-log coordinates distort the graphs of relations and functions because the axes aren t linear Straight lines in Cartesian or rectangular coordinates usually show up as curves in semilog or log-log coordinates Some functions whose graphs appear as curves in Cartesian or rectangular coordinates turn out to be straight lines in semilog or log-log coordinates Try plotting some linear, logarithmic, and exponential functions in Cartesian, semilog, and log-log coordinates See for yourself what happens! Use a calculator, plot numerous points, and then connect the dots for each function you want to graph
Exponential and Logarithmic Curves
Here s a challenge!
Plot graphs of each of the following three functions in x-linear semilog coordinates, y-linear semilog coordinates, and log-log coordinates (use the templates from Figs 14-13 through 14-15): y=x y = ln x y = ex
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