0

_{3}M + 3log_{3}N = 1 + log_{0.008}5

then

which one is true

1) M^{9}=9/N

2) N^{9}=9/M

3) M^{3}=3/N

4) N^{9}=3/M

0

1^{st} let us find the value of log_{0.0008}5.

Let log_{0.0008}5 = y.

So 5 = (0.008)^{y}.

5 = (8/1000)^{y}

5 = (2/10)^{3y}

5 = (1/5)^{3y}

5 = (5)^{-3y}

-3y = 1. y = -1/3.

So, 1 + log_{0.0008}5 = 1 – 1/3 = 2/3.

Now, the options need us to find a relation between M & N, and hence **it should be true for all values of M & N.**

We have the equation :1/3 log_{3}M + 3 log_{3}N = 2/3.

Since it is true for all values of M and N,

Let us split and give:

1/3 log_{3}M = 1/3

3 log_{3}N = 1/3.

(as this also satisfies the equation)

That means log_{3}M = 1, and M = 3.

Also, it means, log_{3}N = 1/9. N = 3^{(1/9)}.

N^{9 }= 3 and M = 3.

So M * N^{9 }= 3*3 = 9.

This is correct as per the 2^{nd} option.