```Question 239105
{{{5^(3-x)= 1/125}}}
There are two keys to this problem:<ul><li>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.</li><li>Either way, we need to recognize that 125 and 1/125 are powers of 5. {{{125 = 5^3}}} and {{{1/125 = 5^(-3)}}}</li></ul>
Using the second key, we can rewrite your equation as:
{{{5^(3-x)= 5^(-3)}}}
Now we can solve the equation without logarithms by recognizing that the only way  {{{5^(3-x)}}} can be the same as {{{5^(-3)}}} is if {{{3-x = -3}}}. So all we have to do is solve
3-x = -3
Subtract 3 from each side:
-x = -6
Divide both sides by -1:
x = 6<br>
We could also solve this with logarithms that we can figure out in our heads. Take
{{{5^(3-x)= 5^(-3)}}}
and find the base 5 logarithms of each side:
{{{log(5, (5^(3-x)))= log(5, (5^(-3)))}}}
Now how do we figure out these logarithms without a calculator? It's not hard if you understand what logarithms represent. {{{log(5, (5^(3-x)))}}} represents "the exponent for 5 which results in {{{5^(3-x)}}}". So what exponent can we put on 5 to get {{{5^(3-x)}}}? Clearly, the answer is 3-x. We can use the same logic to determine that {{{log(5, (5^(-3)))}}} (i.e. the exponent for 5 which results in {{{5^(-3)}}}) is -3. So with these logarithms we determined without calculators we end up with the same equation
3-x = -3
as we got above. And, of course, the solution is the same:
x = 6```