Calculus Examples

$- 2 + 2 \ln (3 x) = 17$

Step 1

Move all terms not containing $\ln (3 x)$ to the right side of the equation.

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Step 1.1

Add $2$ to both sides of the equation.

$2 \ln (3 x) = 17 + 2$

Step 1.2

Add $17$ and $2$ .

$2 \ln (3 x) = 19$

Step 2

Divide each term in $2 \ln (3 x) = 19$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 \ln (3 x) = 19$ by $2$ .

$\frac{2 \ln (3 x)}{2} = \frac{19}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{2 \ln (3 x)}{2} = \frac{19}{2}$

Step 2.2.1.2

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

$\ln (3 x) = \frac{19}{2}$

Step 3

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

$e^{\ln (3 x)} = e^{\frac{19}{2}}$

Step 4

Rewrite $\ln (3 x) = \frac{19}{2}$ 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 $b^{y} = x$ .

$e^{\frac{19}{2}} = 3 x$

Step 5

Solve for $x$ .

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Step 5.1

Rewrite the equation as $3 x = e^{\frac{19}{2}}$ .

$3 x = e^{\frac{19}{2}}$

Step 5.2

Divide each term in $3 x = e^{\frac{19}{2}}$ by $3$ and simplify.

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Step 5.2.1

Divide each term in $3 x = e^{\frac{19}{2}}$ by $3$ .

$\frac{3 x}{3} = \frac{e^{\frac{19}{2}}}{3}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $3$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{e^{\frac{19}{2}}}{3}$

Step 5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{e^{\frac{19}{2}}}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{e^{\frac{19}{2}}}{3}$

Decimal Form:

$x = 4453.24227655 \dots$