Question 1033620: 1/3 log3 x^6 +1/6 log3 x^6 - 1/9 log3x
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! 1/3 * log3(x^6) + 1/6 * log3(x^6) - 1/9 * log3(x) is your expression.
since 1/3 * log3(x^6) is equivalent to log3((x^6)^(1/3)) which is equal to log3(x^2), ...
and since 1/6 * log3(x^6) is equivalent to log3((x^6)^(1/6)) which is equal to log3(x), ...
and since 1/9 * log3(x) is equivalent to log3((x^(1/9)) which is equal to log3(x^(1/9)), your expression becomes ...
log3(x^2) + log3(x) - log3(x^(1/9).
since log3(x^2) + log3(x) is equivalent to log3(x^2*x) which is equal to log3(x^3), your expression becomes ...
log3(x^3) - log3(x^(1/9)).
since log3(x^3) - log3(x^(1/9)) is equivalent to log3(x^3 / x^(1/9)) which is equal to log3(x^(26/9), your expression becomes ...
log3(x^(26/9).
i believe that's as simple as it gets.
you can confirm that your solution is correct by assigning a random value to x and then evaluating the original expression and the final expression to see that they give you the same answer.
i chose x = 5 and i got the same answer for both the original expression and the final expression.
my answer was 4.232145727.
it helps to use the log conversion formula.
log3(x) = log10(x) / log10(3).
this allows you to use your calculator since the calculator log function assumes log10.
using your calculator, you would simply enter log3(x) as log(x) / log(3).
for the final expression, i entered log3(x^(26/9)) as log(5^(26/9)) / log(3).
for the original expression, i entered 1/3 * log3(x^6) + 1/6 * log3(x^6) - 1/9 * log3(x) as (1/3 * log(5^6) + 1/6 * log(5^6) - 1/9 * log(5)) / log(3).
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