Precalculus Examples

$e^{2 x - 1} = {(e^{3})}^{- x}$

Step 1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$e^{2 x - 1} = e^{3 (- x)}$

Step 2

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

$2 x - 1 = 3 (- x)$

Step 3

Solve for $x$ .

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Step 3.1

Multiply $- 1$ by $3$ .

$2 x - 1 = - 3 x$

Step 3.2

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

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Step 3.2.1

Add $3 x$ to both sides of the equation.

$2 x - 1 + 3 x = 0$

Step 3.2.2

Add $2 x$ and $3 x$ .

$5 x - 1 = 0$

Step 3.3

Add $1$ to both sides of the equation.

$5 x = 1$

Step 3.4

Divide each term in $5 x = 1$ by $5$ and simplify.

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Step 3.4.1

Divide each term in $5 x = 1$ by $5$ .

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

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $5$ .

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Step 3.4.2.1.1

Cancel the common factor.

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

Step 3.4.2.1.2

Divide $x$ by $1$ .

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.2$