Precalculus Examples

$\ln (x) - \ln (2) = 0$

Step 1

Use the quotient property of logarithms,

$\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{x}{2}) = 0$

Step 2

To solve for

$x$ , rewrite the equation using properties of logarithms.

$e^{\ln (\frac{x}{2})} = e^{0}$

Step 3

Rewrite

$\ln (\frac{x}{2}) = 0$ 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

$b^{y} = x$ .

$e^{0} = \frac{x}{2}$

Step 4

Solve for

$x$ .

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Step 4.1

Rewrite the equation as

$\frac{x}{2} = e^{0}$ .

$\frac{x}{2} = e^{0}$

Step 4.2

Multiply both sides of the equation by

$2$ .

$2 \frac{x}{2} = 2 e^{0}$

Step 4.3

Simplify both sides of the equation.

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Step 4.3.1

Simplify the left side.

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Step 4.3.1.1

Cancel the common factor of

$2$ .

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Step 4.3.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 e^{0}$

Step 4.3.1.1.2

Rewrite the expression.

$x = 2 e^{0}$

Step 4.3.2

Simplify the right side.

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Step 4.3.2.1

Simplify

$2 e^{0}$ .

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Step 4.3.2.1.1

Anything raised to

$0$ is

$1$ .

$x = 2 \cdot 1$

Step 4.3.2.1.2

Multiply

$2$ by

$1$ .

$x = 2$

$\ln (x) - \ln (2) = 0$

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