Exponential equations quiz worksheet with problems to solve for x, including 3^x = 3^5, 4^x = 4, 5^x = 5^6, 6^x = 6^10, 2^x = 2^10, 6(3^x) = 2^2x, (1/3)^x = 9^x-1, and 8^x = 25.

Problem: Solve for \( x \) in each exponential equation.

We will solve each problem step by step. Let's go through them one by one.

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#### 1. \( 2^x = 2^3 \)

- Both sides of the equation have the same base (\( 2 \)).
- For exponential equations with the same base, the exponents must be equal.
- Therefore, \( x = 3 \).

Solution: \( x = 3 \)

---

#### 2. \( 4^{2x} = 4 \)

- Rewrite \( 4 \) as \( 4^1 \):
\[
4^{2x} = 4^1
\]
- Since the bases are the same, set the exponents equal to each other:
\[
2x = 1
\]
- Solve for \( x \):
\[
x = \frac{1}{2}
\]

Solution: \( x = \frac{1}{2} \)

---

#### 3. \( 5^{1+x} = 5^x \)

- The bases are the same, so set the exponents equal to each other:
\[
1 + x = x
\]
- Subtract \( x \) from both sides:
\[
1 = 0
\]
- This is a contradiction, so there is no solution.

Solution: No solution

---

#### 4. \( 6^x = 6^{2x} \)

- The bases are the same, so set the exponents equal to each other:
\[
x = 2x
\]
- Subtract \( x \) from both sides:
\[
0 = x
\]
- Therefore, \( x = 0 \).

Solution: \( x = 0 \)

---

#### 5. \( 2^x = x^2 \)

- This equation involves different bases, so we cannot directly equate the exponents.
- We need to solve it by inspection or graphing. Let’s test some values of \( x \):
- For \( x = 2 \):
\[
2^2 = 2^2 \quad \text{(True)}
\]
- For \( x = 4 \):
\[
2^4 = 16 \quad \text{and} \quad 4^2 = 16 \quad \text{(True)}
\]
- Testing other values shows that these are the only solutions.

Solution: \( x = 2 \) or \( x = 4 \)

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#### 6. \( (10)^x = 10^{2x} \)

- The bases are the same, so set the exponents equal to each other:
\[
x = 2x
\]
- Subtract \( x \) from both sides:
\[
0 = x
\]
- Therefore, \( x = 0 \).

Solution: \( x = 0 \)

---

#### 7. \( (1/3)^x = 3^{-x} \)

- Rewrite \( (1/3)^x \) as \( 3^{-x} \):
\[
3^{-x} = 3^{-x}
\]
- The equation is true for all \( x \).

Solution: All real numbers \( x \)

---

#### 8. \( 5^x = 25 \)

- Rewrite \( 25 \) as \( 5^2 \):
\[
5^x = 5^2
\]
- Since the bases are the same, set the exponents equal to each other:
\[
x = 2
\]

Solution: \( x = 2 \)

---

Final Answers:

1. \( x = 3 \)
2. \( x = \frac{1}{2} \)
3. No solution
4. \( x = 0 \)
5. \( x = 2 \) or \( x = 4 \)
6. \( x = 0 \)
7. All real numbers \( x \)
8. \( x = 2 \)

\[
\boxed{
\begin{array}{ll}
1. & x = 3 \\
2. & x = \frac{1}{2} \\
3. & \text{No solution} \\
4. & x = 0 \\
5. & x = 2 \text{ or } x = 4 \\
6. & x = 0 \\
7. & \text{All real numbers } x \\
8. & x = 2 \\
\end{array}
}
\]