Precalculus Examples

$4^{5 x} = 16^{2 x - 1}$

Step 1

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

$4^{5 x} = 4^{2 (2 x - 1)}$

Step 2

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

$5 x = 2 (2 x - 1)$

Step 3

Solve for $x$ .

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Step 3.1

Simplify $2 (2 x - 1)$ .

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Step 3.1.1

Apply the distributive property.

$5 x = 2 (2 x) + 2 \cdot - 1$

Step 3.1.2

Multiply.

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Step 3.1.2.1

Multiply $2$ by $2$ .

$5 x = 4 x + 2 \cdot - 1$

Step 3.1.2.2

Multiply $2$ by $- 1$ .

$5 x = 4 x - 2$

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.

$5 x - 4 x = - 2$

Step 3.2.2

Subtract $4 x$ from $5 x$ .

$x = - 2$