## 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 😉