document.write( "Question 239105: solve without using a calculator:
\n" ); document.write( "5^(3-x)= 1/125 \n" ); document.write( "

Algebra.Com's Answer #175621 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"5%5E%283-x%29=+1%2F125\"$
\n" ); document.write( "There are two keys to this problem:

Normally, we would use logarithms to solve an equation where the variable is in the exponent. Since we are asked to solve this without a calculator, either we don't need logarithms or we will be able to figure out the logarithms without using a calculator.
Either way, we need to recognize that 125 and 1/125 are powers of 5. $\"125+=+5%5E3\"$ and $\"1%2F125+=+5%5E%28-3%29\"$

\n" ); document.write( "Using the second key, we can rewrite your equation as:
\n" ); document.write( " $\"5%5E%283-x%29=+5%5E%28-3%29\"$
\n" ); document.write( "Now we can solve the equation without logarithms by recognizing that the only way $\"5%5E%283-x%29\"$ can be the same as $\"5%5E%28-3%29\"$ is if $\"3-x+=+-3\"$ . So all we have to do is solve
\n" ); document.write( "3-x = -3
\n" ); document.write( "Subtract 3 from each side:
\n" ); document.write( "-x = -6
\n" ); document.write( "Divide both sides by -1:
\n" ); document.write( "x = 6

\n" ); document.write( "We could also solve this with logarithms that we can figure out in our heads. Take
\n" ); document.write( " $\"5%5E%283-x%29=+5%5E%28-3%29\"$
\n" ); document.write( "and find the base 5 logarithms of each side:
\n" ); document.write( " $\"log%285%2C+%285%5E%283-x%29%29%29=+log%285%2C+%285%5E%28-3%29%29%29\"$
\n" ); document.write( "Now how do we figure out these logarithms without a calculator? It's not hard if you understand what logarithms represent. $\"log%285%2C+%285%5E%283-x%29%29%29\"$ represents \"the exponent for 5 which results in $\"5%5E%283-x%29\"$ \". So what exponent can we put on 5 to get $\"5%5E%283-x%29\"$ ? Clearly, the answer is 3-x. We can use the same logic to determine that $\"log%285%2C+%285%5E%28-3%29%29%29\"$ (i.e. the exponent for 5 which results in $\"5%5E%28-3%29\"$ ) is -3. So with these logarithms we determined without calculators we end up with the same equation
\n" ); document.write( "3-x = -3
\n" ); document.write( "as we got above. And, of course, the solution is the same:
\n" ); document.write( "x = 6 \n" ); document.write( "

\n" );