Solve for x natural log of 3x- natural log of 2=4

$\ln (3 x) - \ln (2) = 4$

Step 1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{3 x}{2}) = 4$

Step 2

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

$e^{\ln (\frac{3 x}{2})} = e^{4}$

Step 3

Rewrite $\ln (\frac{3 x}{2}) = 4$ 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 $b^{y} = x$ .

$e^{4} = \frac{3 x}{2}$

Step 4

Solve for $x$ .

Rewrite the equation as $\frac{3 x}{2} = e^{4}$ .

$\frac{3 x}{2} = e^{4}$

Multiply both sides of the equation by $\frac{2}{3}$ .

$\frac{2}{3} \cdot \frac{3 x}{2} = \frac{2}{3} e^{4}$

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{2}{3} \cdot \frac{3 x}{2}$ .

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2}{3} \cdot \frac{3 x}{2} = \frac{2}{3} e^{4}$

Rewrite the expression.

$\frac{1}{3} (3 x) = \frac{2}{3} e^{4}$

Cancel the common factor of $3$ .

Factor $3$ out of $3 x$ .

$\frac{1}{3} (3 (x)) = \frac{2}{3} e^{4}$

Cancel the common factor.

$\frac{1}{3} (3 x) = \frac{2}{3} e^{4}$

Rewrite the expression.

$x = \frac{2}{3} e^{4}$

Simplify the right side.

Combine $\frac{2}{3}$ and $e^{4}$ .

$x = \frac{2 e^{4}}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{2 e^{4}}{3}$

Decimal Form:

$x = 36.39876668 \dots$

Solve for x natural log of 3x- natural log of 2=4

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