It looks like you have provided an image with various logarithmic equations. Let's go through each one step by step and solve them.

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

We can use the property of logarithms $\log_b(a) = c$ which means $a = b^c$. In this case, the logarithm is in base 10 (common logarithm).

$\log(2x) = -1 \implies 2x = 10^{-1} = 0.1$ Solving for $x$: $x = \frac{0.1}{2} = 0.05$

c) $\log(3x) = 2$

Again using the property of logarithms: $\log(3x) = 2 \implies 3x = 10^2 = 100$ Solving for $x$: $x = \frac{100}{3} \approx 33.33$

e) $\log x = \frac{3}{2}$

Using the logarithmic property: $\log x = \frac{3}{2} \implies x = 10^{\frac{3}{2}} = \sqrt{10^3} = \sqrt{1000} \approx 31.62$

g) $\log x + \log 50 = \log 100$

Using the logarithmic property $\log a + \log b = \log(ab)$: $\log(x \cdot 50) = \log 100 \implies x \cdot 50 = 100$ Solving for $x$: $x = \frac{100}{50} = 2$

i) $\log x = 1 + \log(22 - x)$

We can isolate the logarithmic terms: $\log x - \log(22 - x) = 1 \implies \log\left(\frac{x}{22 - x}\right) = 1$ This means: $\frac{x}{22 - x} = 10^1 = 10$ Now, solving the equation: $x = 10(22 - x) \implies x = 220 - 10x \implies 11x = 220 \implies x = 20$

k) $5\log(2x) = 20$

Divide both sides by 5: $\log(2x) = 4$ Now, use the logarithmic property: $2x = 10^4 = 10000$ Solving for $x$: $x = \frac{10000}{2} = 5000$

Would you like more details or have any other questions about the steps? Here are some related questions that might help:

1. What are some additional properties of logarithms that are useful for solving equations?
2. How do you solve logarithmic equations with different bases (e.g., natural logarithms)?
3. Can logarithmic equations ever have no solutions?
4. What is the inverse relationship between exponential and logarithmic functions?
5. How do you handle logarithms involving multiple terms on both sides of the equation?

Tip: Always remember that you can combine logarithms on the same side using properties like $\log a + \log b = \log(ab)$ and $\log a - \log b = \log\left(\frac{a}{b}\right)$.