Question
If $27^\text{x}=\frac{9}{3^\text{x}},$ find x.
✓
Answer
We have,
$27^\text{x}=\frac{9}{3^\text{x}}$
$\Rightarrow\big(3^3\big)^\text{x}=\frac{3^2}{3^\text{x}}$
$\Rightarrow3^{3\text{x}}=\frac{3^2}{3^\text{x}}$
$\Rightarrow3^{3\text{x}}\times3^\text{x}=\text{3}^2$
$\Rightarrow3^{3\text{x}}+\text{x}=3^2$
$\Rightarrow3^{4\text{x}}=3^2$
On equating the exponents, we get
$4\text{x}=2$
$\Rightarrow\text{x}=\frac{2}{4}=\frac{1}{2}$
Hence, $\text{x}=\frac{1}{2}$
$27^\text{x}=\frac{9}{3^\text{x}}$
$\Rightarrow\big(3^3\big)^\text{x}=\frac{3^2}{3^\text{x}}$
$\Rightarrow3^{3\text{x}}=\frac{3^2}{3^\text{x}}$
$\Rightarrow3^{3\text{x}}\times3^\text{x}=\text{3}^2$
$\Rightarrow3^{3\text{x}}+\text{x}=3^2$
$\Rightarrow3^{4\text{x}}=3^2$
On equating the exponents, we get
$4\text{x}=2$
$\Rightarrow\text{x}=\frac{2}{4}=\frac{1}{2}$
Hence, $\text{x}=\frac{1}{2}$
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