Precalculus Examples

\ln (x - 4) (\ln (x y^{2}) + 3 \ln (x^{- 1})) + \ln (\frac{y^{6}}{x^{3}})

$\ln (x - 4) (\ln (x y^{2}) + 3 \ln (x^{- 1})) + \ln (\frac{y^{6}}{x^{3}})$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Rewrite the expression using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

\ln (x - 4) (\ln (x y^{2}) + 3 \ln (\frac{1}{x})) + \ln (\frac{y^{6}}{x^{3}})

$\ln (x - 4) (\ln (x y^{2}) + 3 \ln (\frac{1}{x})) + \ln (\frac{y^{6}}{x^{3}})$

Step 1.2

Simplify

3 \ln (\frac{1}{x})

$3 \ln (\frac{1}{x})$ by moving

3

$3$ inside the logarithm.

\ln (x - 4) (\ln (x y^{2}) + \ln ({(\frac{1}{x})}^{3})) + \ln (\frac{y^{6}}{x^{3}})

$\ln (x - 4) (\ln (x y^{2}) + \ln ({(\frac{1}{x})}^{3})) + \ln (\frac{y^{6}}{x^{3}})$

Step 1.3

Apply the product rule to

\frac{1}{x}

$\frac{1}{x}$ .

\ln (x - 4) (\ln (x y^{2}) + \ln (\frac{1^{3}}{x^{3}})) + \ln (\frac{y^{6}}{x^{3}})

$\ln (x - 4) (\ln (x y^{2}) + \ln (\frac{1^{3}}{x^{3}})) + \ln (\frac{y^{6}}{x^{3}})$

Step 1.4

One to any power is one.

\ln (x - 4) (\ln (x y^{2}) + \ln (\frac{1}{x^{3}})) + \ln (\frac{y^{6}}{x^{3}})

$\ln (x - 4) (\ln (x y^{2}) + \ln (\frac{1}{x^{3}})) + \ln (\frac{y^{6}}{x^{3}})$

\ln (x - 4) (\ln (x y^{2}) + \ln (\frac{1}{x^{3}})) + \ln (\frac{y^{6}}{x^{3}})

$\ln (x - 4) (\ln (x y^{2}) + \ln (\frac{1}{x^{3}})) + \ln (\frac{y^{6}}{x^{3}})$

Step 2

Use the product property of logarithms,

\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)

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

\ln (x - 4) \ln (x y^{2} \frac{1}{x^{3}}) + \ln (\frac{y^{6}}{x^{3}})

$\ln (x - 4) \ln (x y^{2} \frac{1}{x^{3}}) + \ln (\frac{y^{6}}{x^{3}})$

Step 3

Cancel the common factor of

x

$x$ .

Tap for more steps...

Step 3.1

Factor

x

$x$ out of

x y^{2}

$x y^{2}$ .

\ln (x - 4) \ln (x (y^{2}) \frac{1}{x^{3}}) + \ln (\frac{y^{6}}{x^{3}})

$\ln (x - 4) \ln (x (y^{2}) \frac{1}{x^{3}}) + \ln (\frac{y^{6}}{x^{3}})$

Step 3.2

Factor

x

$x$ out of

x^{3}

$x^{3}$ .

\ln (x - 4) \ln (x (y^{2}) \frac{1}{x \cdot x^{2}}) + \ln (\frac{y^{6}}{x^{3}})

$\ln (x - 4) \ln (x (y^{2}) \frac{1}{x \cdot x^{2}}) + \ln (\frac{y^{6}}{x^{3}})$

Step 3.3

Cancel the common factor.

\ln (x - 4) \ln (x y^{2} \frac{1}{x \cdot x^{2}}) + \ln (\frac{y^{6}}{x^{3}})

$\ln (x - 4) \ln (x y^{2} \frac{1}{x \cdot x^{2}}) + \ln (\frac{y^{6}}{x^{3}})$

Step 3.4

Rewrite the expression.

\ln (x - 4) \ln (y^{2} \frac{1}{x^{2}}) + \ln (\frac{y^{6}}{x^{3}})

$\ln (x - 4) \ln (y^{2} \frac{1}{x^{2}}) + \ln (\frac{y^{6}}{x^{3}})$

\ln (x - 4) \ln (y^{2} \frac{1}{x^{2}}) + \ln (\frac{y^{6}}{x^{3}})

$\ln (x - 4) \ln (y^{2} \frac{1}{x^{2}}) + \ln (\frac{y^{6}}{x^{3}})$

Step 4

Combine

y^{2}

$y^{2}$ and

\frac{1}{x^{2}}

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

\ln (x - 4) \ln (\frac{y^{2}}{x^{2}}) + \ln (\frac{y^{6}}{x^{3}})

$\ln (x - 4) \ln (\frac{y^{2}}{x^{2}}) + \ln (\frac{y^{6}}{x^{3}})$