Solve for x 3^x*3^(x+7)=9

$3^{x} \cdot 3^{x + 7} = 9$

Step 1

Multiply $3^{x}$ by $3^{x + 7}$ by adding the exponents.

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

$3^{x + x + 7} = 9$

Add $x$ and $x$ .

$3^{2 x + 7} = 9$

Step 2

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

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

Step 3

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

$2 x + 7 = 2$

Step 4

Solve for $x$ .

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

Subtract $7$ from both sides of the equation.

$2 x = 2 - 7$

Subtract $7$ from $2$ .

$2 x = - 5$

Divide each term in $2 x = - 5$ by $2$ and simplify.

Divide each term in $2 x = - 5$ by $2$ .

$\frac{2 x}{2} = \frac{- 5}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 5}{2}$

Divide $x$ by $1$ .

$x = \frac{- 5}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{5}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{5}{2}$

Decimal Form:

$x = - 2.5$

Mixed Number Form:

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