Precalculus Examples

$8^{x - 2} = 16^{x}$

Step 1

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

$2^{3 (x - 2)} = 2^{4 x}$

Step 2

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

$3 (x - 2) = 4 x$

Step 3

Solve for $x$ .

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Step 3.1

Simplify $3 (x - 2)$ .

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Step 3.1.1

Rewrite.

$0 + 0 + 3 (x - 2) = 4 x$

Step 3.1.2

Simplify by adding zeros.

$3 (x - 2) = 4 x$

Step 3.1.3

Apply the distributive property.

$3 x + 3 \cdot - 2 = 4 x$

Step 3.1.4

Multiply $3$ by $- 2$ .

$3 x - 6 = 4 x$

Step 3.2

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

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Step 3.2.1

Subtract $4 x$ from both sides of the equation.

$3 x - 6 - 4 x = 0$

Step 3.2.2

Subtract $4 x$ from $3 x$ .

$- x - 6 = 0$

Step 3.3

Add $6$ to both sides of the equation.

$- x = 6$

Step 3.4

Divide each term in $- x = 6$ by $- 1$ and simplify.

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Step 3.4.1

Divide each term in $- x = 6$ by $- 1$ .

$\frac{- x}{- 1} = \frac{6}{- 1}$

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.

$\frac{x}{1} = \frac{6}{- 1}$

Step 3.4.2.2

Divide $x$ by $1$ .

$x = \frac{6}{- 1}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Divide $6$ by $- 1$ .

$x = - 6$