Logaritm and its properties

Notation: $\log x = \log_{10} x$

We can use log properties to make both equations look nicer. Let's start with the top one: $\begin{aligned} \log 2 + \log y &= 2 \\ \Longrightarrow \log 2xy &= \log 100 \\ \Longrightarrow 2xy &= 100 \end{aligned}$

Now for the second one: $\begin{aligned} \log x^2 - \log 2y &= 4 \\ \Longrightarrow \log \frac{x^2}{2y} &= \log 10^4 \\ \Longrightarrow \frac{x^2}{2y}&=10^4 \end{aligned}$

We can rearrange the top one for $2y$ : $\begin{aligned} 2xy &= 100 \\ \Longrightarrow 2y &= \frac{100}{x} \end{aligned}$

We can plug that into the second equation and solve for $x$ : $\begin{aligned} \frac{x^2}{\blue{2y}} &=10^4 \\ \Longrightarrow \frac{x^2}{\blue{\frac{100}{x}}} &= 10^4 \\ \Longrightarrow \frac{x^3}{100} &= 10^4 \\ \Longrightarrow x^3 &= 10^6 \\ \Longrightarrow x &= \boxed{100} \end{aligned}$

We can use this to solve for $y$ : $\begin{aligned} 2y &= \frac{100}{\blue{x}} \\ \Longrightarrow 2y &= \frac{100}{\blue{100}} = 1 \\ \Longrightarrow y &= \boxed{\frac{1}{2}} \end{aligned}$

Now that we have $x$ and $y$ , we can see that $x + y = \cfrac{\blue{201}}{\orange{2}}$ where $\blue{m=201}$ and $\orange{n=2}$ . Now we can use this to find our answer: $\blue{m} - 3\orange{n}^6 = \blue{201} - 3(\orange{2})^6 = 201 - 192 = \green{\boxed{9}}$

$\begin{cases} \log_{10}(2x) + \log_{10} y = \log_{10} 2 + \log_{10} x + \log_{10} y = 2 & ...(1) \\ \log_{10} x^2 - \log_{10} (2y) = 2 \log_{10} x - \log_{10} 2 - \log_{10} y = 4 &...(2) \end{cases}$

From $(1)+(2)$ :

$\begin{aligned} 3\log_{10} x & = 6 \\ \log_{10} x & = 2 \\ \implies x & = 10^2 = 100 \end{aligned}$

From $(1)$ :

$\begin{aligned} \log_{10} 2 + 2 + \log_{10} y & = 2 \\ \log_{10} y & = - \log_{10} 2 \\ \implies y & = \frac 12 \end{aligned}$

Therefore $x+y = 100 + \dfrac 12 = \dfrac {201}2$ . Then $m - 3n^6 = 201 - 3(2^6) = \boxed 9$ .

log 2x+log y=2
2xy=100 ....... (1)
and log x square-log 2y=4
x square/2y=10000 .......(2)
from 1 and 2
x cube=10 power 6
y=1/2
x+y=100+1/2=201/2
m=201 and n=2
m-3n power 6=201-3(2)^6=201-192=9