Question 164711
solve the indicial equation -- 
<pre><font size = 4 color = "indigo"><b>
{{{5^x = 3^(x+1)}}}

Take the natural log of both sides:

{{{ln(5^x) = ln(3^(x+1))}}}

{{{x*ln(5) = (x+1)ln(3)}}}

To make solving easier,

let {{{A = ln(5)}}} and  {{{B = ln(3)}}}

{{{x*A = (x+1)B}}}

{{{Ax = B(x+1)}}}

{{{Ax = Bx+B}}}

{{{Ax-Bx=B}}}

{{{x(A-B)=B}}}

{{{x=B/(A-B)}}}

Now replace {{{A}}} by {{{ln(5)}}} and {{{B}}} by {{{ln(3)}}}

{{{  x=ln(3)/(ln(5)-ln(3))   }}} 

{{{x=1.098612289/(1.609437912-1.098612289)}}}

{{{x=1.098612289/(.5108256238)}}}

{{{x=2.150660103}}}

Edwin</pre>