Precalculus Examples

$\ln ({(x)}^{6}) - 5 \cdot \ln (x^{\frac{3}{5}})$

Step 1

Simplify each term.

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Step 1.1

Simplify $- 5 \ln (x^{\frac{3}{5}})$ by moving $5$ inside the logarithm.

$\ln (x^{6}) - \ln ({(x^{\frac{3}{5}})}^{5})$

Step 1.2

Multiply the exponents in ${(x^{\frac{3}{5}})}^{5}$ .

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Step 1.2.1

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

$\ln (x^{6}) - \ln (x^{\frac{3}{5} \cdot 5})$

Step 1.2.2

Cancel the common factor of $5$ .

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Step 1.2.2.1

Cancel the common factor.

$\ln (x^{6}) - \ln (x^{\frac{3}{5} \cdot 5})$

Step 1.2.2.2

Rewrite the expression.

$\ln (x^{6}) - \ln (x^{3})$

Step 2

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

$\ln (\frac{x^{6}}{x^{3}})$

Step 3

Cancel the common factor of $x^{6}$ and $x^{3}$ .

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Step 3.1

Factor $x^{3}$ out of $x^{6}$ .

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

Step 3.2

Cancel the common factors.

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Step 3.2.1

Multiply by $1$ .

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 3.2.4

Divide $x^{3}$ by $1$ .

$\ln (x^{3})$