Solve for x 1/9=27^(7x-6)

$\frac{1}{9} = 27^{7 x - 6}$

Step 1

Rewrite the equation as $27^{7 x - 6} = \frac{1}{9}$ .

$27^{7 x - 6} = \frac{1}{9}$

Step 2

Raise $9$ to the power of $1$ .

$27^{7 x - 6} = \frac{1}{9^{1}}$

Step 3

Move $9^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$27^{7 x - 6} = 9^{- 1}$

Step 4

Create equivalent expressions in the equation that all have equal bases.

$3^{3 (7 x - 6)} = 3^{- 2}$

Step 5

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$3 (7 x - 6) = - 2$

Step 6

Solve for $x$ .

Divide each term in $3 (7 x - 6) = - 2$ by $3$ and simplify.

Divide each term in $3 (7 x - 6) = - 2$ by $3$ .

$\frac{3 (7 x - 6)}{3} = \frac{- 2}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 (7 x - 6)}{3} = \frac{- 2}{3}$

Divide $7 x - 6$ by $1$ .

$7 x - 6 = \frac{- 2}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$7 x - 6 = - \frac{2}{3}$

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

Add $6$ to both sides of the equation.

$7 x = - \frac{2}{3} + 6$

To write $6$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$7 x = - \frac{2}{3} + 6 \cdot \frac{3}{3}$

Combine $6$ and $\frac{3}{3}$ .

$7 x = - \frac{2}{3} + \frac{6 \cdot 3}{3}$

Combine the numerators over the common denominator.

$7 x = \frac{- 2 + 6 \cdot 3}{3}$

Simplify the numerator.

Multiply $6$ by $3$ .

$7 x = \frac{- 2 + 18}{3}$

Add $- 2$ and $18$ .

$7 x = \frac{16}{3}$

Divide each term in $7 x = \frac{16}{3}$ by $7$ and simplify.

Divide each term in $7 x = \frac{16}{3}$ by $7$ .

$\frac{7 x}{7} = \frac{\frac{16}{3}}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 x}{7} = \frac{\frac{16}{3}}{7}$

Divide $x$ by $1$ .

$x = \frac{\frac{16}{3}}{7}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{16}{3} \cdot \frac{1}{7}$

Multiply $\frac{16}{3} \cdot \frac{1}{7}$ .

Multiply $\frac{16}{3}$ by $\frac{1}{7}$ .

$x = \frac{16}{3 \cdot 7}$

Multiply $3$ by $7$ .

$x = \frac{16}{21}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{16}{21}$

Decimal Form:

$x = 0.\overline{761904}$

Solve for x 1/9=27^(7x-6)

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