Evaluate 9^(2x)*0.27^(3-x)=1/9

$9^{2 x} \cdot {0.27}^{(3 - x)} = \frac{1}{9}$

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (9^{2 x} \cdot {0.27}^{3 - x}) = \ln (\frac{1}{9})$

Expand the left side.

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Rewrite $\ln (9^{2 x} \cdot {0.27}^{3 - x})$ as $\ln (9^{2 x}) + \ln ({0.27}^{3 - x})$ .

$\ln (9^{2 x}) + \ln ({0.27}^{3 - x}) = \ln (\frac{1}{9})$

Expand $\ln (9^{2 x})$ by moving $2 x$ outside the logarithm.

$2 x \ln (9) + \ln ({0.27}^{3 - x}) = \ln (\frac{1}{9})$

Expand $\ln ({0.27}^{3 - x})$ by moving $3 - x$ outside the logarithm.

$2 x \ln (9) + (3 - x) \ln (0.27) = \ln (\frac{1}{9})$

Apply the distributive property.

$2 x \ln (9) + 3 \ln (0.27) - x \ln (0.27) = \ln (\frac{1}{9})$

Move $3 \ln (0.27)$ .

$2 x \ln (9) - x \ln (0.27) + 3 \ln (0.27) = \ln (\frac{1}{9})$

Move all the terms containing a logarithm to the left side of the equation.

$2 x \ln (9) - x \ln (0.27) + 3 \ln (0.27) - \ln (\frac{1}{9}) = 0$

Move all terms not containing $x$ to the right side of the equation.

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Subtract $3 \ln (0.27)$ from both sides of the equation.

$2 x \ln (9) - x \ln (0.27) - \ln (\frac{1}{9}) = - 3 \ln (0.27)$

Add $\ln (\frac{1}{9})$ to both sides of the equation.

$2 x \ln (9) - x \ln (0.27) = - 3 \ln (0.27) + \ln (\frac{1}{9})$

Factor $x$ out of $2 x \ln (9) - x \ln (0.27)$ .

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Factor $x$ out of $2 x \ln (9)$ .

$x (2 \ln (9)) - x \ln (0.27) = - 3 \ln (0.27) + \ln (\frac{1}{9})$

Factor $x$ out of $- x \ln (0.27)$ .

$x (2 \ln (9)) + x (- 1 \ln (0.27)) = - 3 \ln (0.27) + \ln (\frac{1}{9})$

Factor $x$ out of $x (2 \ln (9)) + x (- 1 \ln (0.27))$ .

$x (2 \ln (9) - 1 \ln (0.27)) = - 3 \ln (0.27) + \ln (\frac{1}{9})$

Rewrite $- 1 \ln (0.27)$ as $- \ln (0.27)$ .

$x (2 \ln (9) - \ln (0.27)) = - 3 \ln (0.27) + \ln (\frac{1}{9})$

Divide each term by $2 \ln (9) - \ln (0.27)$ and simplify.

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Divide each term in $x (2 \ln (9) - \ln (0.27)) = - 3 \ln (0.27) + \ln (\frac{1}{9})$ by $2 \ln (9) - \ln (0.27)$ .

$\frac{x (2 \ln (9) - \ln (0.27))}{2 \ln (9) - \ln (0.27)} = \frac{- 3 \ln (0.27)}{2 \ln (9) - \ln (0.27)} + \frac{\ln (\frac{1}{9})}{2 \ln (9) - \ln (0.27)}$

Cancel the common factor of $2 \ln (9) - \ln (0.27)$ .

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Cancel the common factor.

$\frac{x (2 \ln (9) - \ln (0.27))}{2 \ln (9) - \ln (0.27)} = \frac{- 3 \ln (0.27)}{2 \ln (9) - \ln (0.27)} + \frac{\ln (\frac{1}{9})}{2 \ln (9) - \ln (0.27)}$

Divide $x$ by $1$ .

$x = \frac{- 3 \ln (0.27)}{2 \ln (9) - \ln (0.27)} + \frac{\ln (\frac{1}{9})}{2 \ln (9) - \ln (0.27)}$

Simplify $\frac{- 3 \ln (0.27)}{2 \ln (9) - \ln (0.27)} + \frac{\ln (\frac{1}{9})}{2 \ln (9) - \ln (0.27)}$ .

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Move the negative in front of the fraction.

$x = - \frac{3 \ln (0.27)}{2 \ln (9) - \ln (0.27)} + \frac{\ln (\frac{1}{9})}{2 \ln (9) - \ln (0.27)}$

Simplify terms.

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Combine the numerators over the common denominator.

$x = \frac{- 3 \ln (0.27) + \ln (\frac{1}{9})}{2 \ln (9) - \ln (0.27)}$

Factor $- 1$ out of $- 3 \ln (0.27)$ .

$x = \frac{- (3 \ln (0.27)) + \ln (\frac{1}{9})}{2 \ln (9) - \ln (0.27)}$

Factor $- 1$ out of $\ln (\frac{1}{9})$ .

$x = \frac{- (3 \ln (0.27)) - 1 (- \ln (\frac{1}{9}))}{2 \ln (9) - \ln (0.27)}$

Factor $- 1$ out of $- (3 \ln (0.27)) - 1 (- \ln (\frac{1}{9}))$ .

$x = \frac{- (3 \ln (0.27) - \ln (\frac{1}{9}))}{2 \ln (9) - \ln (0.27)}$

Simplify the expression.

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Rewrite $- (3 \ln (0.27) - \ln (\frac{1}{9}))$ as $- 1 (3 \ln (0.27) - \ln (\frac{1}{9}))$ .

$x = \frac{- 1 (3 \ln (0.27) - \ln (\frac{1}{9}))}{2 \ln (9) - \ln (0.27)}$

Move the negative in front of the fraction.

$x = - \frac{3 \ln (0.27) - \ln (\frac{1}{9})}{2 \ln (9) - \ln (0.27)}$

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{3 \ln (0.27) - \ln (\frac{1}{9})}{2 \ln (9) - \ln (0.27)}$

Decimal Form:

$x = 0.30344344 \dots$

Evaluate 9^(2x)*0.27^(3-x)=1/9

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