Solve for x 7^(-x)*7^(-x)=343

$7^{- x} \cdot 7^{- x} = 343$

Step 1

Multiply $7^{- x}$ by $7^{- x}$ by adding the exponents.

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

$7^{- x - x} = 343$

Subtract $x$ from $- x$ .

$7^{- 2 x} = 343$

Step 2

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

$7^{- 2 x} = 7^{3}$

Step 3

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

$- 2 x = 3$

Step 4

Divide each term in $- 2 x = 3$ by $- 2$ and simplify.

Divide each term in $- 2 x = 3$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{3}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{3}{- 2}$

Divide $x$ by $1$ .

$x = \frac{3}{- 2}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{3}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{3}{2}$

Decimal Form:

$x = - 1.5$

Mixed Number Form:

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