Powharmonic series: Difference between revisions

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An equality involving exponents and logarithms helps us understand why:

<math>\qquad x^{\log_{b}a} = a^{log_{b}x}

<math>\qquad n^{\log_{b}a} = a^{log_{b}n}

</math>

Breaking this down step by step:

# <math>\log_{b}x</math> gives the power to which <math>b</math> must be raised to give <math>x</math>

# <math>\log_{b}n</math> gives the power to which <math>b</math> must be raised to give <math>n</math>

# whenever <math>x</math> is an integer power (squared, cubed, etc.) of <math>b</math>, <math>\log_{b}x</math> will be an integer

# whenever <math>n</math> is an integer power (squared, cubed, etc.) of <math>b</math>, <math>\log_{b}n</math> will be an integer

# whenever <math>\log_{b}x</math> is an integer, we raise <math>a</math> to an integer power

# whenever <math>\log_{b}n</math> is an integer, we raise <math>a</math> to an integer power

# <math>x</math>, being the pitch # or index, increments linearly by 1

# <math>n</math>, being the pitch # or index, increments linearly by 1

# it takes longer and longer each time for <math>x</math> to reach the next power of <math>b</math>

# it takes longer and longer each time for <math>n</math> to reach the next power of <math>b</math>

=== initial count ===

@@ Line 159: / Line 159: @@
 An equality involving exponents and logarithms helps us understand why:
-<math>\qquad x^{\log_{b}a} = a^{log_{b}x}
+<math>\qquad n^{\log_{b}a} = a^{log_{b}n}
 </math>
 Breaking this down step by step:
-# <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>
+# <span><math>\log_{b}n</math></span> gives the power to which <span><math>b</math></span> must be raised to give <span><math>n</math></span>
-# whenever <span><math>x</math></span> is an integer power (squared, cubed, etc.) of <span><math>b</math></span>, <span><math>\log_{b}x</math></span> will be an integer
+# whenever <span><math>n</math></span> is an integer power (squared, cubed, etc.) of <span><math>b</math></span>, <span><math>\log_{b}n</math></span> will be an integer
-# whenever <span><math>\log_{b}x</math></span> is an integer, we raise <span><math>a</math></span> to an integer power
+# whenever <span><math>\log_{b}n</math></span> is an integer, we raise <span><math>a</math></span> to an integer power
-# <span><math>x</math></span>, being the pitch # or index, increments linearly by 1
+# <span><math>n</math></span>, being the pitch # or index, 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>
+# it takes longer and longer each time for <span><math>n</math></span> to reach the next power of <span><math>b</math></span>
 === initial count ===