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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