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

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Question 285384: 25%5Ex=1%2F5
Answer by jsmallt9(3758) About Me  (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, x+=+-1%2F2!

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:
log%285%2C+%2825%5Ex%29%29+=+log%285%2C+%281%2F5%29%29
Using a property of logarithms, log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29, we can move the exponent of the logarithm on the left side out in front:
x%2Alog%285%2C+%2825%29%29+=+log%285%2C+%281%2F5%29%29
Since 25+=+5%5E2 and 1%2F5+=+5%5E%28-1%29, the logarithms are 2 and -1, respectively:
x%2A2+=+-1
Then we divide by 2:
x+=+-1%2F2

Using base 10 logarithms instead:
log%28%2825%5Ex%29%29+=+log%28%281%2F5%29%29
x%2Alog%28%2825%29%29+=+log%28%281%2F5%29%29
x+=+log%28%281%2F5%29%29%2Flog%28%2825%29%29
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.)