Question 1129970
<pre>
Find the exact value of e^LOGe^2 16

The answer is 4, but I am unsure as to why.

If possible, please show multiple methods (with step by step breakdown). Here is what I've gathered so far:

Method 1:
1. e^LOGe^2 2^4
2. e^(4/2)LOGe 2 ( I don't understand where the 4 divided by 2 comes into play, and which property dictates it )
3. e^2LOGe 2
4. e^LOGe 4 (Where did the 4 come from? Did both of the 2's get multiplied?)
5. =4

Method 2:
1. e^LOGe^2 16
2. e^(1/2)LN 16 (Where did the 1/2 come from?)
3. sqrt(e^LN 16) (Why is the entire expression inside the square root?)
4. sqrt(16) (Why did the "e^LN" portion inside the square root disappear?)
5. =4

Is it also possible to solve this problem by using the base change method?

Thanks in advance.
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Find the exact value of e^LOGe^2 16 ===> {{{e^((log (e^2, (16))))}}}
             Let {{{t = log (e^2, (16))}}}
             {{{(e^2)^t = 16}}} ---- Converting to EXPONENTIAL form
             {{{(e^t)^2 = (4)^2}}}
                {{{e^t = 4}}} ----- Exponents are equal. and so are the bases
                 {{{t = log (e, (4))}}}
        {{{highlight(log (e^2, (16))) = highlight(log (e, (4)))}}} ---- Back-substituting {{{log (e^2, (16))}}} for t

APPLYING {{{highlight(a)^((log (highlight(a), (highlight_green(b))))) = highlight_green(b)}}}, {{{matrix(2,1, " ", e^((log (e^2, (16)))))}}} now becomes: {{{matrix(2,1, " ", highlight(e)^((log (highlight(e), (highlight_green(4))))))}}}, which is SIMPLY <font color = green><font size = 4><b>4</font></font></b>
<font color = red><font size = 6><b><u>OR</font></font></b></u>
Find the exact value of e^LOGe^2 16 ===> {{{matrix(2,1, " ", e^((log (e^2, (16)))))}}}
First of all, let's EXTRACT the exponent, {{{log (e^2, (16))}}} from the expression.
{{{highlight(log (e^2, (16))) = (log ((16)))/(log ((e^2))) = (log ((16)))/(2 log (e)) = (1/2)((log ((16)))/(log ((e)))) = (1/2)log (e, (16)) = log (e, (16)^(1/2)) = log (e, (sqrt(16))) = highlight(log (e, (4)))}}}

APPLYING {{{highlight(a)^((log (highlight(a), (highlight_green(b))))) = highlight_green(b)}}}, {{{matrix(2,1, " ", e^((log (e^2, (16)))))}}} now becomes: {{{matrix(2,1, " ", highlight(e)^((log (highlight(e), (highlight_green(4))))))}}}, which is SIMPLY <font color = green><font size = 4><b>4</font></font></b>.</pre>