Trigonometry Examples

$e^{2 x - 1} = {(e^{5})}^{- x}$

Step 1

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

$e^{2 x - 1} = e^{5 (- x)}$

Step 2

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

$2 x - 1 = 5 (- x)$

Step 3

Solve for $x$ .

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Step 3.1

Multiply $- 1$ by $5$ .

$2 x - 1 = - 5 x$

Step 3.2

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

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Step 3.2.1

Add $5 x$ to both sides of the equation.

$2 x - 1 + 5 x = 0$

Step 3.2.2

Add $2 x$ and $5 x$ .

$7 x - 1 = 0$

Step 3.3

Add $1$ to both sides of the equation.

$7 x = 1$

Step 3.4

Divide each term in $7 x = 1$ by $7$ and simplify.

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Step 3.4.1

Divide each term in $7 x = 1$ by $7$ .

$\frac{7 x}{7} = \frac{1}{7}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $7$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{7 x}{7} = \frac{1}{7}$

Step 3.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{7}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{7}$

Decimal Form:

$x = 0.\overline{142857}$