Question: Evaluate the following expression using the properties of logarithms.
log₅(5)+log₅(25)+log₅(625)
○ 7
○ 3
○ 25
○ 15
1. **State the problem:** Evaluate $$\log_5(5) + \log_5(25) + \log_5(625)$$ using properties of logarithms.
2. **Recall the properties:** The logarithm of a product is the sum of the logarithms: $$\log_b(x) + \log_b(y) = \log_b(xy)$$.
3. **Apply the property:** Combine the terms:
$$\log_5(5) + \log_5(25) + \log_5(625) = \log_5(5 \times 25 \times 625)$$
4. **Calculate the product inside the log:**
$$5 \times 25 = 125$$
$$125 \times 625 = 78125$$
So,
$$\log_5(5 \times 25 \times 625) = \log_5(78125)$$
5. **Express 78125 as a power of 5:**
Since $$5^1 = 5$$, $$5^2 = 25$$, $$5^4 = 625$$, let's find the exponent for 78125.
Note that $$78125 = 5^?$$
Calculate:
$$5^7 = 5^{4+2+1} = 5^4 \times 5^2 \times 5^1 = 625 \times 25 \times 5 = 78125$$
6. **Evaluate the logarithm:**
$$\log_5(78125) = 7$$
**Final answer:** $$7$$