Final answer:
To simplify log₆ 6 - 3 log₃ 3 + log₂ 27, we use logarithmic properties to find that log₆ 6 is 1, 3 log₃ 3 is 3, and log₂ 27 simplifies to 3 log₂ 3. Upon combining these values, the expression simplifies to log₂ 3.
Step-by-step explanation:
The task is to simplify the following expression: log₆ 6 - 3 log₃ 3 + log₂ 27.
We start by using the property of logarithms that indicates the logarithm of a number to the base of itself is 1.
Therefore, log₆ 6 simplifies to 1.
Next, we apply the property that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number.
Consequently, 3 log₃ 3 is 3 times 1, which equals 3.
Now, we simplify log₂ 27.
Knowing that 27 is 3 cubed, we can rewrite this as log₂ (3³).
Using the property of logarithms related to exponents, this becomes 3 log₂ 3.
Combining all parts, we get:
Since -3 + 3 equals 0, only log₂ 3 remains.
Thus, the simplified expression is log₂ 3.