SOLVE:
Log (base 5 of the 3rd root of 25)
Third root means to the 1/3 power, so we have Logbase5(25)^1/3.
So the equation we have is Logbase5(25)^1/3
= X, this gives 5^X = 25^1/3. From here we log both sides
and get
Log5^X = Log(25^1/3). Notice now we have gotten rid of the
base five. Using the Power Rule of both sides we have: XLog5
= 1/3Log25, so X = 1/3(Log25)/Log5. But from here you should
notice that Log25 can be broken down into Log5 + Log5 so we
really have:
1/3[(Log5 +Log5)/Log5). After canceling we just have: 1/3(1
+ 1) which is 2/3.
A second shorter way to solve this but maybe harder to see
is that once you have
5^X = 25^1/3 the 25 can be broken down into 5*5 which is
also 5^2 so you really have 5^X equaling (5^2)^1/3 and remember
when you have a power to a power you multiply so 25^1/3 is
really 5^2/3. So you just have 5^X = 5^2/3 so the answer must
be 2/3.
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