I need help, please
find the exact value of
2log3 27+log3 (1/3)-log3 3^1/2
many thanks
(2 * 3) + (-1) - (1/2)
Of course! To find the exact value of the given expression, we can simplify each logarithmic term separately and then combine them.
Let's start with the first term: 2log3 27. This can be rewritten using the logarithmic property that states logb (m^n) = n * logb m. In this case, b is the base 3 and m is 27, so we have:
2log3 27 = 2 * log3 (3^3)
Since 27 is equal to 3^3, we can simplify further:
2 * log3 (3^3) = 2 * log3 3^3 = 2 * 3 = 6
Now let's move on to the second term: log3 (1/3). Using another logarithmic property, logb (1/x) = -logb x, we can rewrite this as:
log3 (1/3) = -log3 3^(-1)
Since 1/3 is equal to 3^(-1), we have:
-log3 3^(-1) = -(-1) = 1
Finally, let's simplify the third term: log3 3^(1/2). Using the property logb (m^a) = a * logb m, we can rewrite this as:
log3 3^(1/2) = (1/2) * log3 3
Since log3 3 is equal to 1 (logarithm with the same base as its argument is always 1), we have:
(1/2) * log3 3 = (1/2) * 1 = 1/2
Now let's put everything together:
2log3 27 + log3 (1/3) - log3 3^(1/2) = 6 + 1 - 1/2
To simplify this expression, we can find a common denominator:
6 + 1 - 1/2 = (12/2) + (2/2) - (1/2) = 15/2
Therefore, the exact value of the given expression is 15/2.