Solve these exponential equations.

Let's solve each of the exponential equations step by step.

---

1. Solve \( (5^2)^3 = 5^x \)

#### Step 1: Simplify the left-hand side
Using the property of exponents \((a^m)^n = a^{m \cdot n}\):
\[
(5^2)^3 = 5^{2 \cdot 3} = 5^6
\]

#### Step 2: Equate the exponents
The equation now becomes:
\[
5^6 = 5^x
\]
Since the bases are the same, we can equate the exponents:
\[
x = 6
\]

#### Final Answer:
\[
\boxed{6}
\]

---

2. Solve \( 3^x = 81 \)

#### Step 1: Express 81 as a power of 3
We know that:
\[
81 = 3^4
\]
So the equation becomes:
\[
3^x = 3^4
\]

#### Step 2: Equate the exponents
Since the bases are the same, we can equate the exponents:
\[
x = 4
\]

#### Final Answer:
\[
\boxed{4}
\]

---

3. Solve \( 7^x = 2401 \)

#### Step 1: Express 2401 as a power of 7
We need to find the power of 7 that equals 2401. Let's calculate:
\[
7^1 = 7, \quad 7^2 = 49, \quad 7^3 = 343, \quad 7^4 = 2401
\]
So:
\[
2401 = 7^4
\]
The equation becomes:
\[
7^x = 7^4
\]

#### Step 2: Equate the exponents
Since the bases are the same, we can equate the exponents:
\[
x = 4
\]

#### Final Answer:
\[
\boxed{4}
\]

---

4. Solve \( 81^x = 9 \)

#### Step 1: Express 81 and 9 as powers of 3
We know that:
\[
81 = 3^4 \quad \text{and} \quad 9 = 3^2
\]
So the equation becomes:
\[
(3^4)^x = 3^2
\]

#### Step 2: Simplify the left-hand side
Using the property of exponents \((a^m)^n = a^{m \cdot n}\):
\[
(3^4)^x = 3^{4x}
\]
The equation now is:
\[
3^{4x} = 3^2
\]

#### Step 3: Equate the exponents
Since the bases are the same, we can equate the exponents:
\[
4x = 2
\]

#### Step 4: Solve for \(x\)
Divide both sides by 4:
\[
x = \frac{2}{4} = \frac{1}{2}
\]

#### Final Answer:
\[
\boxed{\frac{1}{2}}
\]

---

5. Solve \( 32^x = 2 \)

#### Step 1: Express 32 as a power of 2
We know that:
\[
32 = 2^5
\]
So the equation becomes:
\[
(2^5)^x = 2
\]

#### Step 2: Simplify the left-hand side
Using the property of exponents \((a^m)^n = a^{m \cdot n}\):
\[
(2^5)^x = 2^{5x}
\]
The equation now is:
\[
2^{5x} = 2^1
\]

#### Step 3: Equate the exponents
Since the bases are the same, we can equate the exponents:
\[
5x = 1
\]

#### Step 4: Solve for \(x\)
Divide both sides by 5:
\[
x = \frac{1}{5}
\]

#### Final Answer:
\[
\boxed{\frac{1}{5}}
\]

---

Summary of Answers:

1. \( \boxed{6} \)
2. \( \boxed{4} \)
3. \( \boxed{4} \)
4. \( \boxed{\frac{1}{2}} \)
5. \( \boxed{\frac{1}{5}} \)