Question 1183047
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8^x-2 = 2/25, find x
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            Simpler solution is possible.



<pre>
Your starting equation is


    {{{8^(x-2)}}} = {{{2/25}}}.


It is equivalent to


    {{{2^(3*(x-2))}}} = {{{8/100}}}.


Take logarithm base 10 of both sides


    3*(x-2)*log(2) = 3*log(2) - 2


Simplify

    (3x - 6)*log(2) = 3*log(2) - 2

    (3x - 6 - 3)*log(2) = -2

    (3x - 9)*log(2) = -2

     3x - 9 = - {{{2/log((2))}}}

     3x     = - {{{2/log((2))}}} + 9

      x     = - {{{2/(3*log((2)))}}} + 3 = 0.785381  (approximately; rounded).    <U>ANSWER</U>


<U>CHECK</U>.  The left side of the original equation is  {{{8^(0.785381-2)}}} = 0.080000 = {{{8/100}}} = {{{2/25}}}.    ! Correct !
</pre>

Solved.