Question 315465
Given that {{{log(10,(2))}}} is approximately 0.301 and {{{log(10,(3))}}} approx. 0.477, find {{{log(9,(8))}}}.
<pre><b>
Use the change of base formula:

{{{log(OLD_BASE,(X))=log(NEW_BASE,(X))/log(NEW_BASE,(OLD_BASE))}}}

Let the {{{"OLD_BASE"}}} be 9 and the {{{"NEW_BASE"}}} be 10, and X be 8.

Then we have:

{{{log(9,(8))=log(10,(8))/log(10,(9))}}}

Then write 

{{{log(10,(8))}}} as {{{log(10,(2^3))}}} 

and

write {{{log(10,(9))}}} as {{{log(10,(3^2))}}}

Then use the rule {{{log(B,(A^C))=C*log(B,(A))}}}

to write those as

{{{3log(10,(2))}}} 

and

{{{2log(10,(3))}}}

So from start to finish we have:

{{{log(9,(8))=log(10,(8))/log(10,(9))=log(10,(2^3))/log(10,(3^2))=3log(10,(2))/(2log(10,(3)))=(3*0.301)/(2*0.477)=0.903/0.954=0.947}}} 

Edwin</pre>