Trigonometry Examples

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

Step 1

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

$3^{2 (x - 1)} = 3^{3 x}$

Step 2

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

$2 (x - 1) = 3 x$

Step 3

Solve for $x$ .

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Step 3.1

Simplify $2 (x - 1)$ .

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Step 3.1.1

Rewrite.

$0 + 0 + 2 (x - 1) = 3 x$

Step 3.1.2

Simplify by adding zeros.

$2 (x - 1) = 3 x$

Step 3.1.3

Apply the distributive property.

$2 x + 2 \cdot - 1 = 3 x$

Step 3.1.4

Multiply $2$ by $- 1$ .

$2 x - 2 = 3 x$

Step 3.2

Move all terms containing $x$ to the left side of the equation.

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Step 3.2.1

Subtract $3 x$ from both sides of the equation.

$2 x - 2 - 3 x = 0$

Step 3.2.2

Subtract $3 x$ from $2 x$ .

$- x - 2 = 0$

Step 3.3

Add $2$ to both sides of the equation.

$- x = 2$

Step 3.4

Divide each term in $- x = 2$ by $- 1$ and simplify.

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Step 3.4.1

Divide each term in $- x = 2$ by $- 1$ .

$\frac{- x}{- 1} = \frac{2}{- 1}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{2}{- 1}$

Step 3.4.2.2

Divide $x$ by $1$ .

$x = \frac{2}{- 1}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Divide $2$ by $- 1$ .

$x = - 2$