Precalculus Examples

$4 \ln (3 X) = 15$

Step 1

Divide each term in $4 \ln (3 X) = 15$ by $4$ and simplify.

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Step 1.1

Divide each term in $4 \ln (3 X) = 15$ by $4$ .

$\frac{4 \ln (3 X)}{4} = \frac{15}{4}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $4$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{4 \ln (3 X)}{4} = \frac{15}{4}$

Step 1.2.1.2

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

$\ln (3 X) = \frac{15}{4}$

Step 2

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

$e^{\ln (3 X)} = e^{\frac{15}{4}}$

Step 3

Rewrite $\ln (3 X) = \frac{15}{4}$ 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 $b^{y} = x$ .

$e^{\frac{15}{4}} = 3 X$

Step 4

Solve for $X$ .

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Step 4.1

Rewrite the equation as $3 X = e^{\frac{15}{4}}$ .

$3 X = e^{\frac{15}{4}}$

Step 4.2

Divide each term in $3 X = e^{\frac{15}{4}}$ by $3$ and simplify.

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Step 4.2.1

Divide each term in $3 X = e^{\frac{15}{4}}$ by $3$ .

$\frac{3 X}{3} = \frac{e^{\frac{15}{4}}}{3}$

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.

$\frac{3 X}{3} = \frac{e^{\frac{15}{4}}}{3}$

Step 4.2.2.1.2

Divide $X$ by $1$ .

$X = \frac{e^{\frac{15}{4}}}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$X = \frac{e^{\frac{15}{4}}}{3}$

Decimal Form:

$X = 14.17369400 \dots$