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We need to determine the exact value of the logarithm expression log5 0.2

First we will rewrite 0.2 = 2/10

==> log5 0.2 = log5 (2/10) = log 5 (1/5)

Now we will use the logarithm properties to simplify.

We know that log a/b = log a - log b

==> log5 (1/5) = log5 1 - log 5 5

Now we know that loga 1 = 0  and loga a = 1

==> log5 (1/5) = 0 - 1 = -1

Then the exact value of log5 0.2 = -1

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We have to find the value of log(5) 0.2.

0.2 = 1/5 = 5^-1

Now log a^b = b*log a. And log(a) a = 1

log(5) 0.2

=> log (5) [ 1/5]

=> log (5) [ 5^-1]

=> -1* log(5) 5

=> -1*1

=> -1

Therefore the required value of log(5) 0.2 = -1.

Approved by eNotes Editorial Team
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