Precalculus Examples

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

Step 1

Simplify the left side.

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Step 1.1

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

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

Step 1.2

Cancel the common factor of $2$ and $4$ .

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Step 1.2.1

Factor $2$ out of $2 x$ .

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

Step 1.2.2

Cancel the common factors.

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Step 1.2.2.1

Factor $2$ out of $4$ .

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

Step 1.2.2.2

Cancel the common factor.

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

Step 1.2.2.3

Rewrite the expression.

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

Step 2

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

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

Step 3

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

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

Step 4

Solve for $x$ .

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Step 4.1

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

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

Step 4.2

Multiply both sides of the equation by $2$ .

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

Step 4.3

Simplify the left side.

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Step 4.3.1

Cancel the common factor of $2$ .

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Step 4.3.1.1

Cancel the common factor.

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

Step 4.3.1.2

Rewrite the expression.

$x = 2 e^{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = 2 e^{3}$

Decimal Form:

$x = 40.17107384 \dots$