Problem on logarithms

Question

if lg2=x, lg3=y ,then
i) 2/9
ii) 75
iii) 0.0015
Write logarithm base 10 of x and y Please help me to resolve this problem.

For first one I got this, Is this correct?
10^x=2
10^y=3
=10^x / (10^y)^2
=10^x / 10^2y
=10^x-2y

Answer 1

Since you know the logs of 2 and 3, you need to express each question in terms of 2 and 3 (using multiplication, division and exponents).

So, $\frac{2}{9}=\frac{2}{3^2}$ then you can use the logarithm rules to convert that into $\log2-2\log3$ and you can plug in what you know is $\log2$ and $\log3$.

$$\begin{align}\log\frac{2}{9}&=\log\frac{2}{3^2}\\ & =\log2-\log3^2\\ & =\log2-2\log3\\&=x-2y\end{align}$$