If (125)$^x$ = 3125, then the value of x is :

Q: If (125)$^x$ = 3125, then the value of x is :

To solve the equation $ (125)^x = 3125 $ and determine the value of $ x $, we need to express both 125 and 3125 as powers of the same base. 1. **Express 125 as a power of 5:** \[ 125 = 5^3 \] 2. **Express 3125 as a power of 5:** \[ 3125 = 5^5 \] 3. **Substitute these expressions into the original equation:** \[ (5^3)^x = 5^5 \] 4. **Apply the power of a power property, which states that $(a^m)^n = a^{m \cdot n}$:** \[ 5^{3x} = 5^5 \] 5. **Since the bases are the same, we can equate the exponents:** \[ 3x = 5 \] 6. **Solve for $ x $:** \[ x = \frac{5}{3} \] Therefore, the value of $ x $ is $\frac{5}{3}$, which corresponds to Option 4. **Explanation for why other options are incorrect:** - **Option 1: $\frac{3}{2}$** - If $ x = \frac{3}{2} $, then $ 3x = 3 \times \frac{3}{2} = \frac{9}{2} $, which does not equal 5. Therefore, this option is incorrect. - **Option 2: 3** - If $ x = 3 $, then $ 3x = 3 \times 3 = 9 $, which does not equal 5. Therefore, this option is incorrect. - **Option 3: $\frac{3}{5}$** - If $ x = \frac{3}{5} $, then $ 3x = 3 \times \frac{3}{5} = \frac{9}{5} $, which does not equal 5. Therefore, this option is incorrect. In conclusion, the correct value of $ x $ is $\frac{5}{3}$, making Option 4 the correct choice.

Question

Accepted Answer

To solve the equation $ (125)^x = 3125 $ and determine the value of $ x $, we need to express both 125 and 3125 as powers of the same base.

1. **Express 125 as a power of 5:**
   $$
   125 = 5^3
   $$

2. **Express 3125 as a power of 5:**
   $$
   3125 = 5^5
   $$

3. **Substitute these expressions into the original equation:**
   $$
   (5^3)^x = 5^5
   $$

4. **Apply the power of a power property, which states that $(a^m)^n = a^{m \cdot n}$:**
   $$
   5^{3x} = 5^5
   $$

5. **Since the bases are the same, we can equate the exponents:**
   $$
   3x = 5
   $$

6. **Solve for $ x $:**
   $$
   x = \frac{5}{3}
   $$

Therefore, the value of $ x $ is $\frac{5}{3}$, which corresponds to Option 4.

**Explanation for why other options are incorrect:**

- **Option 1: $\frac{3}{2}$**
  - If $ x = \frac{3}{2} $, then $ 3x = 3 \times \frac{3}{2} = \frac{9}{2} $, which does not equal 5. Therefore, this option is incorrect.

- **Option 2: 3**
  - If $ x = 3 $, then $ 3x = 3 \times 3 = 9 $, which does not equal 5. Therefore, this option is incorrect.

- **Option 3: $\frac{3}{5}$**
  - If $ x = \frac{3}{5} $, then $ 3x = 3 \times \frac{3}{5} = \frac{9}{5} $, which does not equal 5. Therefore, this option is incorrect.

In conclusion, the correct value of $ x $ is $\frac{5}{3}$, making Option 4 the correct choice.