Precalculus Examples

$\ln (x^{3}) - 5 \ln (\sqrt[5]{x^{2}})$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Simplify $- 5 \ln (\sqrt[5]{x^{2}})$ by moving $5$ inside the logarithm.

$\ln (x^{3}) - \ln ({\sqrt[5]{x^{2}}}^{5})$

Step 1.2

Rewrite ${\sqrt[5]{x^{2}}}^{5}$ as $x^{2}$ .

Tap for more steps...

Step 1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x^{2}}$ as $x^{\frac{2}{5}}$ .

$\ln (x^{3}) - \ln ({(x^{\frac{2}{5}})}^{5})$

Step 1.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\ln (x^{3}) - \ln (x^{\frac{2}{5} \cdot 5})$

Step 1.2.3

Combine $\frac{2}{5}$ and $5$ .

$\ln (x^{3}) - \ln (x^{\frac{2 \cdot 5}{5}})$

Step 1.2.4

Multiply $2$ by $5$ .

$\ln (x^{3}) - \ln (x^{\frac{10}{5}})$

Step 1.2.5

Cancel the common factor of $10$ and $5$ .

Tap for more steps...

Step 1.2.5.1

Factor $5$ out of $10$ .

$\ln (x^{3}) - \ln (x^{\frac{5 \cdot 2}{5}})$

Step 1.2.5.2

Cancel the common factors.

Tap for more steps...

Step 1.2.5.2.1

Factor $5$ out of $5$ .

$\ln (x^{3}) - \ln (x^{\frac{5 \cdot 2}{5 (1)}})$

Step 1.2.5.2.2

Cancel the common factor.

$\ln (x^{3}) - \ln (x^{\frac{5 \cdot 2}{5 \cdot 1}})$

Step 1.2.5.2.3

Rewrite the expression.

$\ln (x^{3}) - \ln (x^{\frac{2}{1}})$

Step 1.2.5.2.4

Divide $2$ by $1$ .

$\ln (x^{3}) - \ln (x^{2})$

Step 2

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

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

Step 3

Cancel the common factor of $x^{3}$ and $x^{2}$ .

Tap for more steps...

Step 3.1

Factor $x^{2}$ out of $x^{3}$ .

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

Step 3.2

Cancel the common factors.

Tap for more steps...

Step 3.2.1

Multiply by $1$ .

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 3.2.4

Divide $x$ by $1$ .

$\ln (x)$