Precalculus Examples

$4 = \frac{1}{2^{x - 1}}$

Step 1

Rewrite the equation as $\frac{1}{2^{x - 1}} = 4$ .

$\frac{1}{2^{x - 1}} = 4$

Step 2

Move $2^{x - 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$2^{- (x - 1)} = 4$

Step 3

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

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

Step 4

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

$- (x - 1) = 2$

Step 5

Solve for $x$ .

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Step 5.1

Simplify $- (x - 1)$ .

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Step 5.1.1

Apply the distributive property.

$- x - - 1 = 2$

Step 5.1.2

Multiply $- 1$ by $- 1$ .

$- x + 1 = 2$

Step 5.2

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

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Step 5.2.1

Subtract $1$ from both sides of the equation.

$- x = 2 - 1$

Step 5.2.2

Subtract $1$ from $2$ .

$- x = 1$

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Dividing two negative values results in a positive value.

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

Step 5.3.2.2

Divide $x$ by $1$ .

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

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Divide $1$ by $- 1$ .

$x = - 1$