Precalculus Examples

$\frac{119}{e^{6 x} - 14} = 7$

Step 1

Multiply both sides by $e^{6 x} - 14$ .

$\frac{119}{e^{6 x} - 14} (e^{6 x} - 14) = 7 (e^{6 x} - 14)$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Cancel the common factor of $e^{6 x} - 14$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{119}{e^{6 x} - 14} (e^{6 x} - 14) = 7 (e^{6 x} - 14)$

Step 2.1.1.2

Rewrite the expression.

$119 = 7 (e^{6 x} - 14)$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $7 (e^{6 x} - 14)$ .

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Step 2.2.1.1

Apply the distributive property.

$119 = 7 e^{6 x} + 7 \cdot - 14$

Step 2.2.1.2

Multiply $7$ by $- 14$ .

$119 = 7 e^{6 x} - 98$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $7 e^{6 x} - 98 = 119$ .

$7 e^{6 x} - 98 = 119$

Step 3.2

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

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Step 3.2.1

Add $98$ to both sides of the equation.

$7 e^{6 x} = 119 + 98$

Step 3.2.2

Add $119$ and $98$ .

$7 e^{6 x} = 217$

Step 3.3

Divide each term in $7 e^{6 x} = 217$ by $7$ and simplify.

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Step 3.3.1

Divide each term in $7 e^{6 x} = 217$ by $7$ .

$\frac{7 e^{6 x}}{7} = \frac{217}{7}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $7$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{7 e^{6 x}}{7} = \frac{217}{7}$

Step 3.3.2.1.2

Divide $e^{6 x}$ by $1$ .

$e^{6 x} = \frac{217}{7}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $217$ by $7$ .

$e^{6 x} = 31$

Step 3.4

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{6 x}) = \ln (31)$

Step 3.5

Expand the left side.

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Step 3.5.1

Expand $\ln (e^{6 x})$ by moving $6 x$ outside the logarithm.

$6 x \ln (e) = \ln (31)$

Step 3.5.2

The natural logarithm of $e$ is $1$ .

$6 x \cdot 1 = \ln (31)$

Step 3.5.3

Multiply $6$ by $1$ .

$6 x = \ln (31)$

Step 3.6

Divide each term in $6 x = \ln (31)$ by $6$ and simplify.

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Step 3.6.1

Divide each term in $6 x = \ln (31)$ by $6$ .

$\frac{6 x}{6} = \frac{\ln (31)}{6}$

Step 3.6.2

Simplify the left side.

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Step 3.6.2.1

Cancel the common factor of $6$ .

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Step 3.6.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{\ln (31)}{6}$

Step 3.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (31)}{6}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (31)}{6}$

Decimal Form:

$x = 0.57233120 \dots$