Precalculus Examples

$e^{2 x} > 37$

Step 1

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

$\ln (e^{2 x}) > \ln (37)$

Step 2

Expand the left side.

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Step 2.1

Expand $\ln (e^{2 x})$ by moving $2 x$ outside the logarithm.

$2 x \ln (e) > \ln (37)$

Step 2.2

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

$2 x \cdot 1 > \ln (37)$

Step 2.3

Multiply $2$ by $1$ .

$2 x > \ln (37)$

Step 3

Divide each term in $2 x > \ln (37)$ by $2$ and simplify.

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Step 3.1

Divide each term in $2 x > \ln (37)$ by $2$ .

$\frac{2 x}{2} > \frac{\ln (37)}{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} > \frac{\ln (37)}{2}$

Step 3.2.1.2

Divide $x$ by $1$ .

$x > \frac{\ln (37)}{2}$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$x > \frac{\ln (37)}{2}$

Interval Notation:

$(\frac{\ln (37)}{2}, \infty)$

Step 5