Question
Given: $\log_3m = x$ and $\log_3n = y.$Express $3^{2x - 3}$ in terms of $m.$
✓
Answer
Given that $\log _3 m=x$ and $\log _3 n=y$
$\Rightarrow 3^{\mathrm{x}}=\mathrm{m}$ and $3^{\mathrm{y}}=\mathrm{n}$
Consider the given expression :
$3^{2 x-3}$
$ =3^{2 x} \cdot 3^{-3}$
$ =3^{2 x} \cdot \frac{1}{3^3}$
$=\frac{3^{2 x}}{3^3}$
$=\frac{\left(3^x\right)^2}{3^3}$
$=\frac{m^2}{27}$
Therefore, $3^{2 x-3}=\frac{m^2}{27}$
$\Rightarrow 3^{\mathrm{x}}=\mathrm{m}$ and $3^{\mathrm{y}}=\mathrm{n}$
Consider the given expression :
$3^{2 x-3}$
$ =3^{2 x} \cdot 3^{-3}$
$ =3^{2 x} \cdot \frac{1}{3^3}$
$=\frac{3^{2 x}}{3^3}$
$=\frac{\left(3^x\right)^2}{3^3}$
$=\frac{m^2}{27}$
Therefore, $3^{2 x-3}=\frac{m^2}{27}$
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