Use the limit to find the exact values of k when a = 2 and a = 10.

Section 114. Since 0.0 = 1, when the exponent on 0. increases, the power itself increases, provided a. is greater than 1. It follows that if the exponent is infinitely small and positive, then the power also exceeds 1 by an infinitely small number. Let 0: be an infinitely small number, a." = 1 + 1,!) where 2,0 is also an infinitely small number. we let 1,!) 2 km. Then we have a" = 1 + km, and with a. as the base for logarithms, we have 0; = log (1 + kw). EXAMPLE In order that it may be clearer how the number it depends on a, let a = 10. From the table of common logarithms,l we look for the logarithm of a number which exceeds 1 by the smallest l 1 1000000 ' 1000000 ' so that kw = possible amount, for instance, 1 + Then 10g (1 + 1003000) = log $33333 = 000000043429 = 00. Since kw = 000000100000. it follows that i = $350 and k = 133932090 = 2.30258. We see that k is a finite number which depends on the value of the base a. If a different base had been chosen. then the logarithm of the same number 1 + kw will differ from the logarithm already given. It follows that a different value of is will result