Simplify the following expressions: i) log 125 ii) 81a3b5c2 / 27b2c. Choose the correct option for: i) 2 log 5 + log 4 - 2 log 10 ii) 125^(-1/3) 10^(-1) 102

$$\log 125 = \log 5^3 = 3 \log 5$$

The simplified expression is $\boxed{3 \log 5}$.

ii) $81a^3b^5c^2 \div 27b^2c$

$$\frac{81}{27} \cdot a^3 \cdot b^{5-2} \cdot c^{2-1}$$ $$3a^3b^3c^1$$

The simplified expression is $\boxed{3a^3b^3c}$.

3. Choose the correct in

i) $2 \log 5 + \log 4 - 2 \log 10$

$$\log 5^2 + \log 4 - \log 10^2$$ $$\log 25 + \log 4 - \log 100$$

$$\log (25 \times 4) - \log 100$$ $$\log 100 - \log 100$$ $$\log \left(\frac{100}{100}\right)$$ $$\log 1$$

The correct option is A. $\log 1$.

ii) $125^{-\frac{1}{3}} \times 10^{-1} \times 10^2$

$a^{-n} = \frac{1}{a^n}$ and $a^{\frac{1}{n}} = \sqrt[n]{a}$.

$$125^{-\frac{1}{3}} = \frac{1}{125^{\frac{1}{3}}} = \frac{1}{\sqrt[3]{125}} = \frac{1}{5}$$ $$10^{-1} = \frac{1}{10}$$ $$10^2 = 100$$

$$\frac{1}{5} \times \frac{1}{10} \times 100$$ $$\frac{100}{50}$$ $$2$$

It seems there might be a typo in the options provided in the image, as none of them match the calculated answer of 2. Assuming there was a typo and the options were meant to include 2 or if the question intended a different calculation, based on the provided options (A. $1/5$, B. 350, C. $1/50$, D. 1), none are correct. If the question was $125^{-\frac{1}{3}} \times 10^{-1} \times 10^1$, the answer would be $1/5$. Given the options, there might be an error in the question or the options. However, if we strictly follow the calculation, the result is 2.

Let's re-evaluate the options provided in the image. The options are A. $1/5$, B. 350, C. $1/50$, D. 1.
My calculation gives 2. Since 2 is not among the options, I will state the calculated answer.
The simplified expression is $\boxed{2}$.

4. Evaluate $\frac{1}{2} \log 9$

$$\log \sqrt{9}$$ $$\log 3$$

The evaluated expression is $\boxed{\log 3}$.

iii) Write the log form $5^x = 125$

If $b^y = x$, then $\log_b x = y$.

$$5^x = 125 \implies \log_5 125 = x$$

The logarithmic form is $\boxed{\log_5 125 = x}$.

5. Simplify indice
$(\frac{2}{3})^{x-1} = \frac{27}{8}$

$$\left(\frac{2}{3}\right)^{x-1} = \left(\frac{2}{3}\right)^{-3}$$ $$x-1 = -3$$

$$x = -3 + 1$$ $$x = -2$$

The value of $x$ is $\boxed{-2}$.