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THE SCIENCE SITE
Understanding Logarithms III
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Graph of y = log(x) Over Four Decades of Range

Example 6: The above plot is a graph of the rough estimate numbers, y = log(N). The lowest point on the graph is: x = 0.01, y = log (x) = -2. The next point is 2 x= 0.02, for which [y = log (0.02) = [log (0.01) - log(2)] = (-2 + 0.301) = -1.699]. Similarly, [log(0.04) = log(2 x 0.02) = log (0.02) - log(2) = -1.699 + 0.301 = -1.398]. Then log(0.08) = -1.398 +0.301 = -1.097, and log(0.10) = -1.0.

For the second decade, the logarithmic values are equal to those of the first decade plus 1.0. The next two decades are calculated in a similar manner, and all log values are now positive. You will now have 16 guess numbers, although you only need the closest above estimate number to make a determination. Now we have all of the points necessary to calculate any point on the graph, using the method outlined on the previous two pages.

There will, however, be errors when calculation of values near the midpoints on the above curve. In order to reduce the error we will add rough estimate points at the midpoints by simply adding or subtracting (0.301/2) = 0.1505 to the logarithmic values and multiplying or dividing the guess number by sqrt(2). For instance, for N = 2, the next lowest refined estimate point is sqrt(2) = 1.414, and log(1.414) = (0 + 0.1505) = 0.1505 = estimated value, while the actual value is 0.15045.

For Q & A regarding logarithms, see next page

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