Precalculus Examples

$\ln (\sqrt{x}) + \frac{\ln (x)}{\ln (y)} + \ln (\frac{y}{x}) + \frac{1}{2} \cdot \ln (x) + \ln (\frac{1}{y})$

Step 1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\ln (\sqrt{x} \frac{y}{x}) + \frac{\ln (x)}{\ln (y)} + \frac{1}{2} \cdot \ln (x) + \ln (\frac{1}{y})$

Step 2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\frac{\ln (x)}{\ln (y)} + \frac{1}{2} \cdot \ln (x) + \ln (\sqrt{x} \frac{y}{x} \frac{1}{y})$

Step 3

Simplify each term.

Tap for more steps...

Step 3.1

Simplify $\frac{1}{2} \ln (x)$ by moving $\frac{1}{2}$ inside the logarithm.

$\frac{\ln (x)}{\ln (y)} + \ln (x^{\frac{1}{2}}) + \ln (\sqrt{x} \frac{y}{x} \frac{1}{y})$

Step 3.2

Combine $\sqrt{x}$ and $\frac{y}{x}$ .

$\frac{\ln (x)}{\ln (y)} + \ln (x^{\frac{1}{2}}) + \ln (\frac{\sqrt{x} y}{x} \cdot \frac{1}{y})$

Step 3.3

Cancel the common factor of $y$ .

Tap for more steps...

Step 3.3.1

Factor $y$ out of $\sqrt{x} y$ .

$\frac{\ln (x)}{\ln (y)} + \ln (x^{\frac{1}{2}}) + \ln (\frac{y \sqrt{x}}{x} \cdot \frac{1}{y})$

Step 3.3.2

Cancel the common factor.

$\frac{\ln (x)}{\ln (y)} + \ln (x^{\frac{1}{2}}) + \ln (\frac{y \sqrt{x}}{x} \cdot \frac{1}{y})$

Step 3.3.3

Rewrite the expression.

$\frac{\ln (x)}{\ln (y)} + \ln (x^{\frac{1}{2}}) + \ln (\frac{\sqrt{x}}{x})$

Step 4

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\frac{\ln (x)}{\ln (y)} + \ln (x^{\frac{1}{2}} \frac{\sqrt{x}}{x})$

Step 5

Simplify each term.

Tap for more steps...

Step 5.1

Multiply $x^{\frac{1}{2}} \frac{\sqrt{x}}{x}$ .

Tap for more steps...

Step 5.1.1

Combine $x^{\frac{1}{2}}$ and $\frac{\sqrt{x}}{x}$ .

$\frac{\ln (x)}{\ln (y)} + \ln (\frac{x^{\frac{1}{2}} \sqrt{x}}{x})$

Step 5.1.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\frac{\ln (x)}{\ln (y)} + \ln (\frac{x^{\frac{1}{2}} x^{\frac{1}{2}}}{x})$

Step 5.1.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\ln (x)}{\ln (y)} + \ln (\frac{x^{\frac{1}{2} + \frac{1}{2}}}{x})$

Step 5.1.4

Combine the numerators over the common denominator.

$\frac{\ln (x)}{\ln (y)} + \ln (\frac{x^{\frac{1 + 1}{2}}}{x})$

Step 5.1.5

Add $1$ and $1$ .

$\frac{\ln (x)}{\ln (y)} + \ln (\frac{x^{\frac{2}{2}}}{x})$

Step 5.1.6

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.1.6.1

Cancel the common factor.

$\frac{\ln (x)}{\ln (y)} + \ln (\frac{x^{\frac{2}{2}}}{x})$

Step 5.1.6.2

Rewrite the expression.

$\frac{\ln (x)}{\ln (y)} + \ln (\frac{x^{1}}{x})$

Step 5.2

Cancel the common factor of $x^{1}$ and $x$ .

Tap for more steps...

Step 5.2.1

Factor $x$ out of $x^{1}$ .

$\frac{\ln (x)}{\ln (y)} + \ln (\frac{x \cdot 1}{x})$

Step 5.2.2

Cancel the common factors.

Tap for more steps...

Step 5.2.2.1

Raise $x$ to the power of $1$ .

$\frac{\ln (x)}{\ln (y)} + \ln (\frac{x \cdot 1}{x^{1}})$

Step 5.2.2.2

Factor $x$ out of $x^{1}$ .

$\frac{\ln (x)}{\ln (y)} + \ln (\frac{x \cdot 1}{x \cdot 1})$

Step 5.2.2.3

Cancel the common factor.

$\frac{\ln (x)}{\ln (y)} + \ln (\frac{x \cdot 1}{x \cdot 1})$

Step 5.2.2.4

Rewrite the expression.

$\frac{\ln (x)}{\ln (y)} + \ln (\frac{1}{1})$

Step 5.2.2.5

Rewrite the expression.

$\frac{\ln (x)}{\ln (y)} + \ln (1)$

Step 5.3

The natural logarithm of $1$ is $0$ .

$\frac{\ln (x)}{\ln (y)} + 0$

Step 6

Add $\frac{\ln (x)}{\ln (y)}$ and $0$ .

$\frac{\ln (x)}{\ln (y)}$