Precalculus Examples

$\ln (6) - \ln (2 x) = 3$

Step 1

Simplify the left side.

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Step 1.1

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

$\ln (\frac{6}{2 x}) = 3$

Step 1.2

Cancel the common factor of $6$ and $2$ .

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Step 1.2.1

Factor $2$ out of $6$ .

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

Step 1.2.2

Cancel the common factors.

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Step 1.2.2.1

Factor $2$ out of $2 x$ .

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

Step 1.2.2.2

Cancel the common factor.

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

Step 1.2.2.3

Rewrite the expression.

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

Step 2

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

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

Step 3

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

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

Step 4

Solve for $x$ .

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Step 4.1

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

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

Step 4.2

Find the LCD of the terms in the equation.

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Step 4.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x, 1$

Step 4.2.2

The LCM of one and any expression is the expression.

$x$

Step 4.3

Multiply each term in $\frac{3}{x} = e^{3}$ by $x$ to eliminate the fractions.

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Step 4.3.1

Multiply each term in $\frac{3}{x} = e^{3}$ by $x$ .

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

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $x$ .

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Step 4.3.2.1.1

Cancel the common factor.

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

Step 4.3.2.1.2

Rewrite the expression.

$3 = e^{3} x$

Step 4.4

Solve the equation.

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Step 4.4.1

Rewrite the equation as $e^{3} x = 3$ .

$e^{3} x = 3$

Step 4.4.2

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

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Step 4.4.2.1

Divide each term in $e^{3} x = 3$ by $e^{3}$ .

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

Step 4.4.2.2

Simplify the left side.

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Step 4.4.2.2.1

Cancel the common factor of $e^{3}$ .

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Step 4.4.2.2.1.1

Cancel the common factor.

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

Step 4.4.2.2.1.2

Divide $x$ by $1$ .

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.14936120 \dots$