Solve for x (1/8)^(x-5)=4^x

${(\frac{1}{8})}^{x - 5} = 4^{x}$

Step 1

Apply the product rule to $\frac{1}{8}$ .

$\frac{1^{x - 5}}{8^{x - 5}} = 4^{x}$

Step 2

One to any power is one.

$\frac{1}{8^{x - 5}} = 4^{x}$

Step 3

Move $8^{x - 5}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$8^{- (x - 5)} = 4^{x}$

Step 4

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

$2^{3 (- (x - 5))} = 2^{2 x}$

Step 5

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

$3 (- (x - 5)) = 2 x$

Step 6

Solve for $x$ .

Simplify $3 (- (x - 5))$ .

Start simplifying.

$0 + 0 + 3 (- (x - 5)) = 2 x$

Simplify by adding zeros.

$3 (- (x - 5)) = 2 x$

Apply the distributive property.

$3 (- x - - 5) = 2 x$

Multiply $- 1$ by $- 5$ .

$3 (- x + 5) = 2 x$

Apply the distributive property.

$3 (- x) + 3 \cdot 5 = 2 x$

Multiply.

Multiply $- 1$ by $3$ .

$- 3 x + 3 \cdot 5 = 2 x$

Multiply $3$ by $5$ .

$- 3 x + 15 = 2 x$

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

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

$- 3 x + 15 - 2 x = 0$

Subtract $2 x$ from $- 3 x$ .

$- 5 x + 15 = 0$

Subtract $15$ from both sides of the equation.

$- 5 x = - 15$

Divide each term in $- 5 x = - 15$ by $- 5$ and simplify.

Divide each term in $- 5 x = - 15$ by $- 5$ .

$\frac{- 5 x}{- 5} = \frac{- 15}{- 5}$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 x}{- 5} = \frac{- 15}{- 5}$

Divide $x$ by $1$ .

$x = \frac{- 15}{- 5}$

Simplify the right side.

Divide $- 15$ by $- 5$ .

$x = 3$

Solve for x (1/8)^(x-5)=4^x

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