Let's go through each question one by one:

Problem 42

Given: $\log_a (x - 2) + \log_a (2x + 2) = \log_a (5x)$

To solve for $x$:

1. Use the property $\log_a b + \log_a c = \log_a (bc)$. $\log_a \left((x - 2)(2x + 2)\right) = \log_a (5x)$
2. Since the logs are equal, set the arguments equal to each other: $(x - 2)(2x + 2) = 5x$
3. Expand and simplify this equation to solve for $x$.

Problem 43

Given: $3^{2x - 2} - 3^{x + 1} = 3^x - 3^3$

Rewrite each term to express the equation in terms of powers of $3$:

1. Use substitution or logarithmic techniques to simplify and solve for $x$.

Problem 44

Solve the following logarithmic equations:

(a) $\log 3x - \log (2x + 2) = \log (x - 1)$

1. Use the property $\log_a b - \log_a c = \log_a \left(\frac{b}{c}\right)$. $\log \left(\frac{3x}{2x + 2}\right) = \log (x - 1)$
2. Set the arguments equal to each other: $\frac{3x}{2x + 2} = x - 1$
3. Solve this rational equation for $x$.

(b) $\log_3 x + 21(\log_x 3) = 10$

1. Use the change of base property or substitution to express this equation in a simpler form.
2. Solve for $x$.

Problem 45

Solve the following equations:

(a) $3^{x+1} + 3^{2x} = 10$

1. Let $y = 3^x$, so $3^{x+1} = 3 \cdot y$ and $3^{2x} = y^2$.
2. Substitute and form a quadratic equation in terms of $y$, then solve for $y$ and back-substitute to find $x$.

(b) $\sqrt{3x + 1} - \sqrt{x - 4} = 3$

1. Isolate one of the square roots and square both sides to remove the square root.
2. Simplify and solve the resulting equation for $x$.

Would you like detailed steps for any specific question?

Related Questions for Further Practice

1. How would you handle logarithmic equations with different bases?
2. What strategies are effective for solving exponential equations involving multiple powers?
3. How can properties of logarithms simplify complex logarithmic expressions?
4. What are common pitfalls when solving equations involving square roots?
5. How do substitutions simplify exponential equations?

Tip

When dealing with logarithmic equations, always consider rewriting using log properties to combine or simplify terms before isolating variables.