I am ask to express logarithms in terms of a and b

Question

Here the question,I have try to obtain the answer. My answer is $2a+2b-2$, but I not sure with my answer

Given $\log_x2=a$ and $\log_x 3=b$, write $\log_x\dfrac{4}{9x^2}$ in terms of $a$ and $b$

Answer 1

You did a sign mistake. Note $$\log_x\frac{4}{9x^2}=\log_x 4-\log_x(9x^2)\\ =2\log_x 2-(\log_x 9+\log_x x^2)\\= 2a-(2b+2)\\=2a-2b-2$$

Answer 2

$\log_x(2) = a, \log_x(3)=b$. Then

$$\log_x(\frac{4}{9x^2}) = \log_x(4) - \log_x(9x^2) = \log_x(2^2) - \left(\log_x(3^2) + \log_x(x^2)\right)\\ = 2\log_x(2)-2\log_x(3) - 2$$

So we get $2a-2b-2$ after using the definitions of $a$ and $b$. By definition $\log_x(x^2)=2$ of course.

Answer 3

$$N=\log_x\frac{4}{9x^2}=\log_x 4-\log_x 9-\log_x x^2=2\log_x 2-2 \log_x 3-2=2a-2b-2$$