Question 304781
Express {{{e^(log(4,x))}}} as a power of x. 

{{{e^(log(4,x))}}}
<pre><font size = 4 color = "indigo"><b>
Use the change of base formula, where NEW and OLD refer to the
NEW and OLD bases:

{{{log(OLD, A) = log(NEW,A)/log(NEW,OLD)}}}

We want to use that to change {{{log(4,x)}}} from OLD base 4 
to NEW base e:

{{{log(4,x)=log(e,x)/log(e,4)}}}

And we realize that {{{log(e,"")}}} is the same as {{{"ln"}}}

So

{{{log(4,x)=ln(x)/ln(4)}}} 

And therefore

{{{e^(log(4,x))}}}

becomes

{{{drawing(70,75,0,1,-1,1,locate(0,0,e^(ln(x)/ln(4))) )}}}

We write that quotient exponent {{{ln(x)/ln(4)}}} as a product {{{ln(x)*""}}}{{{1/ln(4)}}}

{{{drawing(100,75,0,1,-1,1,locate(0,0,e^(ln(x))*""^(1/ln(4))) )}}}

Now we use the rule {{{B^(A*C) = (B^A)^C}}}

{{{drawing(100,100,0,1,-1,1,locate(0,0,(e^(ln(x)))^(1/ln(4)))) )}}}

Next we use the rule {{{e^(ln(A))=A}}} to write the above as

{{{drawing(100,100,0,1,-1,1,locate(0,0,

x^(1/ln(4))))}}}

and that is a power of x.

Edwin</pre></b></font>