Powharmonic series: Difference between revisions

@@ Line 148: / Line 148: @@
 <math>3, 9, 27, 81…
-</math>An equality involving exponents and logarithms helps us understand why:
+</math>
+An equality involving exponents and logarithms helps us understand why:
 <math>\qquad x^{\log_{b}a} = a^{log_{b}x}
 </math>
-so if log b of x is what power b must be raised to to give x, then as you increment x linearly by 1, whenever you reach a power of b, b will be a natural number, which when you raise a to that, you'll get a "natural"/pure/whatever power of a.
+* <span><math>\log_{b}x</math></span> gives the power to which <span><math>b</math></span> must be raised to give <span><math>x</math></span>
+* whenever <span><math>x</math></span> is a power of <span><math>b</math></span>, <span><math>\log_{b}x</math></span> will be an integer
+* whenever <span><math>\log_{b}x</math></span> is an integer, we raise <span><math>x</math></span> to an integer power
+* <span><math>x</math></span> increments linearly by 1
+* it takes longer and longer each time for <span><math>x</math></span> to reach the next power of <span><math>b</math></span>
 == ln-of-a-powharmonic series ==