Precalculus Examples

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

Step 1

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

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

Step 2

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

$3 x = 2 - x$

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

Add $x$ to both sides of the equation.

$3 x + x = 2$

Step 3.1.2

Add $3 x$ and $x$ .

$4 x = 2$

Step 3.2

Divide each term in $4 x = 2$ by $4$ and simplify.

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Step 3.2.1

Divide each term in $4 x = 2$ by $4$ .

$\frac{4 x}{4} = \frac{2}{4}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $4$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{2}{4}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{4}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Cancel the common factor of $2$ and $4$ .

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Step 3.2.3.1.1

Factor $2$ out of $2$ .

$x = \frac{2 (1)}{4}$

Step 3.2.3.1.2

Cancel the common factors.

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Step 3.2.3.1.2.1

Factor $2$ out of $4$ .

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

Step 3.2.3.1.2.2

Cancel the common factor.

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

Step 3.2.3.1.2.3

Rewrite the expression.

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.5$