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Let's solve these exponential equations one by one:

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

Since both sides have the same base, we can equate the exponents:

\[
x - 1 = 1 \implies x = 2
\]

### 2. \( 17^{5x} = 1 \)

We know that \( 17^0 = 1 \), so:

\[
5x = 0 \implies x = 0
\]

### 3. \( 5^{4x-1} = 25 \)

We can rewrite 25 as \( 5^2 \), so:

\[
5^{4x-1} = 5^2
\]

Now equate the exponents:

\[
4x - 1 = 2 \implies 4x = 3 \implies x = \frac{3}{4}
\]

### 4. \( 16^{2x-1} = 64 \)

Rewrite 16 and 64 as powers of 2: \( 16 = 2^4 \) and \( 64 = 2^6 \). So the equation becomes:

\[
(2^4)^{2x-1} = 2^6 \implies 2^{4(2x-1)} = 2^6
\]

Now equate the exponents:

\[
4(2x-1) = 6 \implies 8x - 4 = 6 \implies 8x = 10 \implies x = \frac{5}{4}
\]

### 5. \( \left( \frac{1}{2} \right)^{x+4} = \left( \frac{2}{3} \right)^{1 - 3x} \)

We cannot directly equate the exponents, so let's take the logarithm of both sides and solve for \( x \).

### 6. \( \left( \frac{1}{7} \right)^{4x} = 7 \)

Rewrite \( 7 \) as \( 7^1 \), and \( \frac{1}{7} \) as \( 7^{-1} \), so the equation becomes:

\[
7^{-4x} = 7^1
\]

Now equate the exponents:

\[
-4x = 1 \implies x = -\frac{1}{4}
\]

### 7. \( \left( \frac{1}{3} \right)^{x^2 - 4x + 3} = 3^{1-x} \)

Rewrite \( \frac{1}{3} \) as \( 3^{-1} \), so:

\[
3^{-(x^2 - 4x + 3)} = 3^{1-x}
\]

Now equate the exponents:

\[
-(x^2 - 4x + 3) = 1 - x
\]

Simplify and solve the quadratic equation:

\[
-x^2 + 4x - 3 = 1 - x
\]
\[
-x^2 + 4x - 3 = 1 - x \implies -x^2 + 5x - 4 = 0
\]

Multiply through by -1:

\[
x^2 - 5x + 4 = 0
\]

Factor the quadratic:

\[
(x - 4)(x - 1) = 0
\]

So, \( x = 4 \) or \( x = 1 \).

### 8. \( 4^{x+1} + 7 \cdot 2^x - 2 = 0 \)

Let's rewrite \( 4^{x+1} \) as \( (2^2)^{x+1} = 2^{2x+2} \):

\[
2^{2x+2} + 7 \cdot 2^x - 2 = 0
\]

Let \( y = 2^x \), so the equation becomes:

\[
y^2 \cdot 4 + 7y - 2 = 0
\]

Now solve this quadratic equation for \( y \), then back-substitute to find \( x \).

---

Would you like details for any specific steps, or do you want help with a particular problem?

Here are some related questions:
1. How can logarithms help in solving exponential equations with different bases?
2. What are the common properties of exponents used in simplifying equations?
3. How do we solve quadratic equations using factoring or the quadratic formula?
4. What is the method to deal with fractional bases in exponential equations?
5. Why is back-substitution important when solving equations with substitutions?

**Tip:** Always check if both sides of an equation can be expressed as powers of the same base. It simplifies solving exponentially.

This problem set includes solving various exponential equations such as 3^(x-1) = 3, 17^(5x) = 1, 5^(4x-1) = 25, and more. Detailed step-by-step solutions covering the use of properties of exponents and logarithmic identities. Ideal for Grade 11 students.

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