Solve for x e^(5x+2)=e^(3x+12)

$e^{5 x + 2} = e^{3 x + 12}$

Step 1

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

$e^{5 x + 2} = e^{3 x + 12}$

Step 2

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

$5 x + 2 = 3 x + 12$

Step 3

Solve for $x$ .

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

Subtract $3 x$ from both sides of the equation.

$5 x + 2 - 3 x = 12$

Subtract $3 x$ from $5 x$ .

$2 x + 2 = 12$

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

Subtract $2$ from both sides of the equation.

$2 x = 12 - 2$

Subtract $2$ from $12$ .

$2 x = 10$

Divide each term in $2 x = 10$ by $2$ and simplify.

Divide each term in $2 x = 10$ by $2$ .

$\frac{2 x}{2} = \frac{10}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{10}{2}$

Divide $x$ by $1$ .

$x = \frac{10}{2}$

Simplify the right side.

Divide $10$ by $2$ .

$x = 5$