SOLUTION: solve without using a calculator: 5^(3-x)= 1/125

Question 239105: solve without using a calculator:
5^(3-x)= 1/125
Found 2 solutions by Theo, jsmallt9:
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
5^(3-x) = 1/125

since 5^3 = 125, then 5^(-3) = 1/5^3 = 1/125

we have 5^-3 = 1/125 which means that:

3-x = -3 which means that
x = 6

your equation becomes:

5^(3-6) = 1/125 which becomes:

5^(-3) = 1/125 which becomes:

1/5^3 = 1/125 which becomes:

1/125 = 1/125 which is true so the answer of x = 6 is good.

Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

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. and

Using the second key, we can rewrite your equation as:

Now we can solve the equation without logarithms by recognizing that the only way can be the same as is if . 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

We could also solve this with logarithms that we can figure out in our heads. Take

and find the base 5 logarithms of each side:

Now how do we figure out these logarithms without a calculator? It's not hard if you understand what logarithms represent. represents "the exponent for 5 which results in ". So what exponent can we put on 5 to get ? Clearly, the answer is 3-x. We can use the same logic to determine that (i.e. the exponent for 5 which results in ) 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
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