Question 461940
{{{log(a,(x))/log(a/b,(x))}}}
<pre>
Using the change of base formula on the numerator only to
natural logs:

{{{NUMERATOR =log(a,(x))=ln(x)/ln(a)}}}

Using the change of base formula on the denominator only to
natural logs: [Note: you could use logs base 10 if you prefer]

{{{DENOMINATOR = log(a/b,(x))=ln(x)/ln(a/b)}}}

So we have

{{{log(a,(x))/log(a/b,(x))}}}{{{""=""}}}{{{NUMERATOR}}}{{{"÷"}}}{{{DENOMINATOR}}}{{{""=""}}}{{{ln(x)/ln(a)}}}{{{"÷"}}}{{{ln(x)/ln(a/b)}}}{{{""=""}}}{{{ln(x)/ln(a)}}}{{{""*""}}}{{{ln(a/b)/ln(x)}}}{{{""=""}}}{{{cross(ln(x))/ln(a)}}}{{{""*""}}}{{{ln(a/b)/cross(ln(x))}}}{{{""=""}}}{{{ln(a/b)/ln(a)}}}{{{""=""}}}{{{(ln(a) - ln(b))/ln(a)}}}{{{""=""}}}{{{ln(a)/ln(a)-ln(b)/ln(a) = 1-ln(b)/ln(a)}}}

If you prefer to use logs base 10 then just replace 
all the {{{"ln(_)"}}}'s with {{{"log(_)"}}}'s and the answer will be
{{{1-"log(b)"/"log(a)"}}}.

Edwin</pre>