1. Express and calculate...

Hi,

If $\log_9 x = a$ and $\log_3 y = b$, express $xy$ and
$\frac{x}{y}$ as powers of $3$.
If $xy = 243$
and $\frac{x}{y} = 3$, calculate $a$ and $b$.

I got the expression part but not the calculation:

$\log_9 x = a,\,\,\,\,\,\,\,\,\,\,\log_3 y = b$

$x = 9^a = 3^{2a}\,\,\,\,\,\,\,\,y = 3^b$

$xy = 3^{2a} \times 3^b$
$xy = 3^{2a + b}$

$\frac{x}{y} = \frac{3^{2a}}{3^b}$
$= 3^{2a - b}$

2. What don't you get? It looks fine.

3. $xy=243=3^5$

$xy=3^{2a+b}$

$\therefore 3^5=3^{2a+b}$...............(I)

likewise, $\frac{x}{y}=3 \;and\;\frac{x}{y}=3^{2a-b}$

$\therefore 3^{1}=3^{2a-b}$................(II)

you can find a and b from equations (I) and (II)

4. Originally Posted by harish21
$xy=243=3^5$

$xy=3^{2a+b}$

$\therefore 3^5=3^{2a+b}$...............(I)

likewise, $\frac{x}{y}=3 \;and\;\frac{x}{y}=3^{2a-b}$

$\therefore 3^{1}=3^{2a-b}$................(II)

you can find a and b from equations (I) and (II)
Ah, I made an attempt like that but I had no faith in it. Thanks.

I'm not sure if this works:

$2a + b = 5$
$2a - b = 1$

$4a = 6$
$a = \frac{3}{2}$

$2(\frac{3}{2}) - b = 1$
$3 - b = 1$
$b = 2$

5. -Well Done!

You can check whether your values for a and b are correct or not by plugging their values back in equations (I) and (II)

Check:

Equation (II)

$3^1=3^{2a-b}$

$3=3^{2\cdot\frac{3}{2}-2}$

$3=3$