Why Logarithms Taper Off

Why do logarithms taper off like they do?

A few posts ago we looked at exponential equations, such as 2^x = y where the 2 is a base.

We all know that for example, 2^3 = 8. A general form is b^x = y where b is the base. Well, with logarithms, the format is log_b y = x. So for 2^3 = 8, we would express that with logarithms as log_2 8=3.

Another way of writing logarithms is switching the x and the y, so the logarithm of b^y = x where b is the base would be log_b x = y. Now, say the base b is 2, and we graph log_b x = y. So here is a table showing the values for log_b x = y for x=1 through x=60. Basically, think of the tapering off this way: when you graph b^y = x for increasing y values, for instance 2^y = x, as y increases more and more, for each subsequent y, the difference between each x becomes increasingly large; the second derivative is positive. Just as for 2^y = x each subsequent y created a larger difference between each subsequent x, conversely, when you switch that around to log_b x = y, for log_b x = y, each subsequent x creates a smaller and smaller difference between each subsequent y.

x base y x base y
1 2 0 31 2 4.954196
2 2 1 32 2 5
3 2 1.585 33 2 5.044394
4 2 2 34 2 5.087463
5 2 2.322 35 2 5.129283
6 2 2.585 36 2 5.169925
7 2 2.807 37 2 5.209453
8 2 3 38 2 5.247928
9 2 3.17 39 2 5.285402
10 2 3.322 40 2 5.321928
11 2 3.459 41 2 5.357552
12 2 3.585 42 2 5.392317
13 2 3.7 43 2 5.426265
14 2 3.807 44 2 5.459432
15 2 3.907 45 2 5.491853
16 2 4 46 2 5.523562
17 2 4.087 47 2 5.554589
18 2 4.17 48 2 5.584963
19 2 4.248 49 2 5.61471
20 2 4.322 50 2 5.643856
21 2 4.392 51 2 5.672425
22 2 4.459 52 2 5.70044
23 2 4.524 53 2 5.72792
24 2 4.585 54 2 5.754888
25 2 4.644 55 2 5.78136
26 2 4.7 56 2 5.807355
27 2 4.755 57 2 5.83289
28 2 4.807 58 2 5.857981
29 2 4.858 59 2 5.882643
30 2 4.907 60 2 5.906891

Disclaimer: I’m no math pro so this could all be wrong 😉

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