Question: Evaluate the following expression using the properties of logarithms.
log₆(6)+log₆(36)+log₆(216)
1. **State the problem:** Evaluate $$\log_6(6) + \log_6(36) + \log_6(216)$$ using properties of logarithms.
2. **Recall the logarithm property:** $$\log_b(x) + \log_b(y) = \log_b(xy)$$ for the same base $b$.
3. **Apply the property:**
$$\log_6(6) + \log_6(36) + \log_6(216) = \log_6(6 \times 36 \times 216)$$
4. **Calculate the product inside the logarithm:**
$$6 \times 36 = 216$$
$$216 \times 216 = 46656$$
5. **Rewrite the expression:**
$$\log_6(46656)$$
6. **Express 46656 as a power of 6:**
Since $$6^1 = 6$$, $$6^2 = 36$$, $$6^3 = 216$$, and $$6^6 = 46656$$ (because $$6^6 = (6^3)^2 = 216^2 = 46656$$).
7. **Use the logarithm definition:**
$$\log_6(6^6) = 6$$
**Final answer:** $$6$$