Precalculus Examples

$7 \ln (7 x) = 14$

Step 1

Divide each term in $7 \ln (7 x) = 14$ by $7$ and simplify.

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Step 1.1

Divide each term in $7 \ln (7 x) = 14$ by $7$ .

$\frac{7 \ln (7 x)}{7} = \frac{14}{7}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $7$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{7 \ln (7 x)}{7} = \frac{14}{7}$

Step 1.2.1.2

Divide $\ln (7 x)$ by $1$ .

$\ln (7 x) = \frac{14}{7}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $14$ by $7$ .

$\ln (7 x) = 2$

Step 2

To solve for $x$ , rewrite the equation using properties of logarithms.

$e^{\ln (7 x)} = e^{2}$

Step 3

Rewrite $\ln (7 x) = 2$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{2} = 7 x$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $7 x = e^{2}$ .

$7 x = e^{2}$

Step 4.2

Divide each term in $7 x = e^{2}$ by $7$ and simplify.

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Step 4.2.1

Divide each term in $7 x = e^{2}$ by $7$ .

$\frac{7 x}{7} = \frac{e^{2}}{7}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $7$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{7 x}{7} = \frac{e^{2}}{7}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{e^{2}}{7}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{e^{2}}{7}$

Decimal Form:

$x = 1.05557944 \dots$