Algebra Examples

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

Step 1

Rewrite $4$ as $2^{2}$ .

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

Step 2

Multiply the exponents in ${(2^{2})}^{x - 4}$ .

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Step 2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.2

Apply the distributive property.

$4^{x - 1} = 2 \cdot 2^{2 x + 2 \cdot - 4}$

Step 2.3

Multiply $2$ by $- 4$ .

$4^{x - 1} = 2 \cdot 2^{2 x - 8}$

Step 3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$4^{x - 1} = 2^{1 + 2 x - 8}$

Step 4

Subtract $8$ from $1$ .

$4^{x - 1} = 2^{2 x - 7}$

Step 5

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

$2^{2 (x - 1)} = 2^{2 x - 7}$

Step 6

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

$2 (x - 1) = 2 x - 7$

Step 7

Solve for $x$ .

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Step 7.1

Simplify $2 (x - 1)$ .

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Step 7.1.1

Rewrite.

$0 + 0 + 2 (x - 1) = 2 x - 7$

Step 7.1.2

Simplify by adding zeros.

$2 (x - 1) = 2 x - 7$

Step 7.1.3

Apply the distributive property.

$2 x + 2 \cdot - 1 = 2 x - 7$

Step 7.1.4

Multiply $2$ by $- 1$ .

$2 x - 2 = 2 x - 7$

Step 7.2

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

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Step 7.2.1

Subtract $2 x$ from both sides of the equation.

$2 x - 2 - 2 x = - 7$

Step 7.2.2

Combine the opposite terms in $2 x - 2 - 2 x$ .

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Step 7.2.2.1

Subtract $2 x$ from $2 x$ .

$0 - 2 = - 7$

Step 7.2.2.2

Subtract $2$ from $0$ .

$- 2 = - 7$

Step 7.3

Since $- 2 \neq - 7$ , there are no solutions.

No solution