<p>There was a somewhat difficult question on the review of logarithm test today.</p>
<p>You're given the graph of 5^x and it goes through a rectangle, intersecting the rectangle at the upper right vertex. At the bottom left vertex of the rectangle is labeled 2 and the bottom right vertex is labeled "a". The area of the rectangle is 500, find "a". </p>
<p>So it took me a second to find out I had to solve the equation 500 = (a-2)(5^a)</p>
<p>Does anyone have a solution that's mostly algebraic with only one final calculation that needs to be put in the calculator? I approximated the answer, but couldn't solve the problem using any other methods.</p>
<p>Just to sanity check, the four vertices of the rectangle are (2, 5^a), (a, 5^a), (2, 0), and (a, 0)? You didn't specify where the fourth side of the rectangle was, but your equation implies that the fourth side is on the x-axis.</p>
<p>If so, you won't find an answer that's mostly algebraic. This is because the only way to get the a out of the exponent would involve taking a logarithm (either common or natural, but you'll find that you prefer natural logarithms in calculus). Taking that logarithm would place the (a-2) as the argument of a logarithm, and the only way to get that out would be to exponentiate... which would leave you back where you started.*</p>
<p>Given that, the most effective way to solve this problem with a calculator would be to set Y1 = (a-2)(5^a) and Y2 = 500, and then to use the "intersect" feature ([2nd][Trace][#5] on the TI-83/84 series) to get an answer that's accurate to three decimal places.</p>
<p>Hope that helps.</p>
<p>*[It should be noted for accuracy that there is a mathematical term for this, but the name of it is currently escaping me, and is well beyond the scope of the typical Calculus class (and is probably beyond the knowledge of most Calculus teachers.]</p>