Solve for x 5=(1/5)^x

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

Step 1

Rewrite the equation as ${(\frac{1}{5})}^{x} = 5$ .

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

Step 2

Apply the product rule to $\frac{1}{5}$ .

$\frac{1^{x}}{5^{x}} = 5$

Step 3

One to any power is one.

$\frac{1}{5^{x}} = 5$

Step 4

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

$5^{- x} = 5$

Step 5

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

$5^{- x} = 5^{1}$

Step 6

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

$- x = 1$

Step 7

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

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

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $1$ by $- 1$ .

$x = - 1$