Question: Solve the logarithmic equation.
log₆x + log₆10 = 1
x =
1. **State the problem:** Solve the logarithmic equation $$\log_6 x + \log_6 10 = 1$$ for $$x$$.
2. **Recall the logarithm property:** The sum of logarithms with the same base can be combined as the logarithm of the product:
$$\log_b A + \log_b B = \log_b (A \times B)$$
3. **Apply the property:**
$$\log_6 x + \log_6 10 = \log_6 (10x)$$
So the equation becomes:
$$\log_6 (10x) = 1$$
4. **Rewrite the logarithmic equation in exponential form:**
$$\log_6 (10x) = 1 \implies 10x = 6^1$$
5. **Simplify the right side:**
$$10x = 6$$
6. **Solve for $$x$$:**
$$x = \frac{6}{10}$$
7. **Simplify the fraction:**
$$x = \frac{3}{5}$$
**Final answer:**
$$x = \frac{3}{5}$$