SOLUTION: {{{25^x=1/5}}}

Question 285384:
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
If you understand exponents well you can just look at this and know what x must be. 1/5 is the reciprocal of 5 and 5 is the square root of 25. Since the exponent for reciprocals is -1 and the exponent for square roots is 1/2, !

If you don't get exponents that well, you can use logarithms. Any base of logarithm will do but the best ones would be 5 (because both 25 and 1/5 are powers of 5), 10 or e (because your calculator "knows" these bases.

Using base 5 logarithms:

Using a property of logarithms, , we can move the exponent of the logarithm on the left side out in front:

Since and , the logarithms are 2 and -1, respectively:

Then we divide by 2:

Using base 10 logarithms instead:

Then we get out our calculator and find the two logarithms and then divide them. You should get -0.5 (which is -1/2 in decimal form) or some number very, very close to -0.5. (Calculators use decimal approximations for logarithms so there may be a small rounding error. This is why it is preferable to avoid calculators when you can.)
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