We need to simplify three logarithmic expressions: i) $log_10 5 + 2 log_10 4$ ii) $2 log 7 - log 14$ iii) $log_5 36 + 2 log_5 7 - \frac{1}{2} log_5 12$

1. Problem Description

We need to simplify three logarithmic expressions:

i) $log_10 5 + 2 log_10 4$

ii) $2 log 7 - log 14$

iii) $log_5 36 + 2 log_5 7 - \frac{1}{2} log_5 12$

2. Solution Steps

i) $log_10 5 + 2 log_10 4$

We use the power rule for logarithms: $n log_a x = log_a (x^n)$.

$2 log_10 4 = log_10 (4^2) = log_10 16$

So, $log_10 5 + log_10 16$

Now, we use the product rule for logarithms: $log_a x + log_a y = log_a (xy)$.

$log_10 5 + log_10 16 = log_10 (5 \cdot 16) = log_10 80$

ii) $2 log 7 - log 14$

Using the power rule for logarithms: $2 log 7 = log (7^2) = log 49$.

So, $log 49 - log 14$.

Now, we use the quotient rule for logarithms: $log_a x - log_a y = log_a (\frac{x}{y})$.

$log 49 - log 14 = log (\frac{49}{14}) = log (\frac{7}{2})$

iii) $log_5 36 + 2 log_5 7 - \frac{1}{2} log_5 12$

Using the power rule for logarithms: $2 log_5 7 = log_5 (7^2) = log_5 49$ and $\frac{1}{2} log_5 12 = log_5 (12^{\frac{1}{2}}) = log_5 \sqrt{12} = log_5 (2\sqrt{3})$.

So, $log_5 36 + log_5 49 - log_5 (2\sqrt{3})$.

Using the product rule for logarithms: $log_5 36 + log_5 49 = log_5 (36 \cdot 49) = log_5 1764$.

Now, $log_5 1764 - log_5 (2\sqrt{3}) = log_5 (\frac{1764}{2\sqrt{3}}) = log_5 (\frac{882}{\sqrt{3}}) = log_5 (\frac{882\sqrt{3}}{3}) = log_5 (294\sqrt{3})$.

3. Final Answer

i) $log_{10} 80$

ii) $log(\frac{7}{2})$

iii) $log_5(294\sqrt{3})$