Final answer:
The expression 6log6(36) simplifies to 12, using the logarithmic property that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number, combined with the fact that log base b of b is always 1.
Step-by-step explanation:
To simplify 6log6(36) using logarithmic properties, we should identify that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number. In this case, the base of the logarithm and the base of the number inside the logarithm are the same, which simplifies the expression significantly.
Firstly, 36 is a power of 6, since 36 = 62. So, we can write the expression as 6log6(62). Applying the logarithmic property that logb(an) = n · logb(a), we get 6 · 2 · log6(6). Since logb(b) is always 1, the expression further simplifies to 6 · 2 · 1, which equals 12.
Therefore, 6log6(36) simplifies to 12.