Precalculus Examples

$5 \ln (3 x) = 14$

Step 1

Divide each term in $5 \ln (3 x) = 14$ by $5$ and simplify.

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Step 1.1

Divide each term in $5 \ln (3 x) = 14$ by $5$ .

$\frac{5 \ln (3 x)}{5} = \frac{14}{5}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $5$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{5 \ln (3 x)}{5} = \frac{14}{5}$

Step 1.2.1.2

Divide $\ln (3 x)$ by $1$ .

$\ln (3 x) = \frac{14}{5}$

Step 2

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

$e^{\ln (3 x)} = e^{\frac{14}{5}}$

Step 3

Rewrite $\ln (3 x) = \frac{14}{5}$ 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 $b^{y} = x$ .

$e^{\frac{14}{5}} = 3 x$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $3 x = e^{\frac{14}{5}}$ .

$3 x = e^{\frac{14}{5}}$

Step 4.2

Divide each term in $3 x = e^{\frac{14}{5}}$ by $3$ and simplify.

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Step 4.2.1

Divide each term in $3 x = e^{\frac{14}{5}}$ by $3$ .

$\frac{3 x}{3} = \frac{e^{\frac{14}{5}}}{3}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $3$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{e^{\frac{14}{5}}}{3}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{e^{\frac{14}{5}}}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{e^{\frac{14}{5}}}{3}$

Decimal Form:

$x = 5.48154892 \dots$