Solve for x 8^(x-1)=(1/2)^(2x-1)

$8^{x - 1} = {(\frac{1}{2})}^{2 x - 1}$

Step 1

Apply the product rule to $\frac{1}{2}$ .

$8^{x - 1} = \frac{1^{2 x - 1}}{2^{2 x - 1}}$

Step 2

One to any power is one.

$8^{x - 1} = \frac{1}{2^{2 x - 1}}$

Step 3

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

$8^{x - 1} = 2^{- (2 x - 1)}$

Step 4

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

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

Step 5

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

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

Step 6

Solve for $x$ .

Simplify $3 (x - 1)$ .

Rewrite.

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

Simplify by adding zeros.

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

Apply the distributive property.

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

Multiply $3$ by $- 1$ .

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

Simplify $- (2 x - 1)$ .

Apply the distributive property.

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

Multiply.

Multiply $2$ by $- 1$ .

$3 x - 3 = - 2 x - - 1$

Multiply $- 1$ by $- 1$ .

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

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

Add $2 x$ to both sides of the equation.

$3 x - 3 + 2 x = 1$

Add $3 x$ and $2 x$ .

$5 x - 3 = 1$

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

Add $3$ to both sides of the equation.

$5 x = 1 + 3$

Add $1$ and $3$ .

$5 x = 4$

Divide each term in $5 x = 4$ by $5$ and simplify.

Divide each term in $5 x = 4$ by $5$ .

$\frac{5 x}{5} = \frac{4}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{4}{5}$

Divide $x$ by $1$ .

$x = \frac{4}{5}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{4}{5}$

Decimal Form:

$x = 0.8$

Solve for x 8^(x-1)=(1/2)^(2x-1)

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