Let’s solve the equations from the uploaded image step by step.

(a) $\log_3(2x - 1) = -2$

1. Rewrite the logarithmic equation in exponential form: $2x - 1 = 3^{-2}$
2. Simplify $3^{-2}$: $2x - 1 = \frac{1}{9}$
3. Solve for $x$: $2x = \frac{1}{9} + 1 = \frac{1}{9} + \frac{9}{9} = \frac{10}{9}$ $x = \frac{10}{18} = \frac{5}{9}$

Solution: $x = \frac{5}{9}$

(b) $5^{\log_5(x)} = 3$

1. Simplify $5^{\log_5(x)}$ to $x$ since $a^{\log_a(b)} = b$: $x = 3$

Solution: $x = 3$

(c) $\log_2(x) - \log_2(3) = 4$

1. Use the logarithmic property $\log_a(m) - \log_a(n) = \log_a\left(\frac{m}{n}\right)$: $\log_2\left(\frac{x}{3}\right) = 4$
2. Rewrite in exponential form: $\frac{x}{3} = 2^4$
3. Solve for $x$: $x = 3 \cdot 16 = 48$

Solution: $x = 48$

(d) $\log_3(5) = x$

1. This is already in logarithmic form. $x$ is simply: $x = \log_3(5)$

Solution: $x = \log_3(5)$ (leave as is unless approximation is needed)

Do you want detailed explanations for any of the steps?

Related Questions:

1. How do you simplify logarithmic expressions with multiple terms?
2. What are the rules for converting logarithmic to exponential form?
3. Can you explain the general property $a^{\log_a(b)} = b$?
4. How do you handle logarithms with bases other than 10 or $e$?
5. What are some real-world applications of logarithmic equations?

Tip:

When solving logarithmic equations, always check that the solution does not result in taking the logarithm of a non-positive number (logarithms are undefined for non-positive arguments).