Let's solve these logarithmic equations one by one.

1. $\ln x = -3$

To solve this, we rewrite the natural logarithm in exponential form:

$x = e^{-3}$

Thus, $x = \frac{1}{e^3} \approx 0.0498$.

2. $\log(3x - 2) = 2$

Rewrite in exponential form using base 10:

$3x - 2 = 10^2 = 100$

Now, solve for $x$:

$3x = 100 + 2 = 102 \quad \Rightarrow \quad x = \frac{102}{3} = 34$

3. $2 \log x = \log 2 + \log(3x - 4)$

Use the properties of logarithms to simplify:

$\log x^2 = \log(2(3x - 4))$

Since the logarithms are equal, we can set the arguments equal:

$x^2 = 2(3x - 4)$

Expanding and solving the quadratic equation:

$x^2 = 6x - 8 \quad \Rightarrow \quad x^2 - 6x + 8 = 0$

Using the quadratic formula:

$x = \frac{6 \pm \sqrt{(-6)^2 - 4(1)(8)}}{2(1)} = \frac{6 \pm \sqrt{36 - 32}}{2} = \frac{6 \pm \sqrt{4}}{2} = \frac{6 \pm 2}{2}$

So, $x = \frac{6 + 2}{2} = 4$ or $x = \frac{6 - 2}{2} = 2$.

Thus, $x = 4$ or $x = 2$.

4. $\log x + \log(x - 1) = \log(4x)$

Combine the logarithms:

$\log(x(x - 1)) = \log(4x)$

Since the logarithms are equal, set the arguments equal:

$x(x - 1) = 4x$

Expand and simplify:

$x^2 - x = 4x \quad \Rightarrow \quad x^2 - 5x = 0 \quad \Rightarrow \quad x(x - 5) = 0$

Thus, $x = 0$ or $x = 5$. However, $x = 0$ is not a valid solution for a logarithmic equation, so the only solution is $x = 5$.

5. $\log_3(x + 25) - \log_3(x - 1) = 3$

Use the properties of logarithms:

$\log_3\left(\frac{x + 25}{x - 1}\right) = 3$

Rewrite in exponential form:

$\frac{x + 25}{x - 1} = 3^3 = 27$

Now solve for $x$:

$x + 25 = 27(x - 1) \quad \Rightarrow \quad x + 25 = 27x - 27 \quad \Rightarrow \quad 25 + 27 = 26x \quad \Rightarrow \quad x = \frac{52}{26} = 2$

6. $\log_9(x - 5) + \log_9(x + 3) = 1$

Combine the logarithms:

$\log_9((x - 5)(x + 3)) = 1$

Rewrite in exponential form:

$(x - 5)(x + 3) = 9^1 = 9$

Expand the quadratic:

$x^2 - 5x + 3x - 15 = 9 \quad \Rightarrow \quad x^2 - 2x - 15 = 9 \quad \Rightarrow \quad x^2 - 2x - 24 = 0$

Solve using the quadratic formula:

$x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-24)}}{2(1)} = \frac{2 \pm \sqrt{4 + 96}}{2} = \frac{2 \pm \sqrt{100}}{2} = \frac{2 \pm 10}{2}$

Thus, (x = \frac{2 + 10}{2}