Precalculus Examples

$\frac{1}{3} \cdot (5 - \ln (x) - \ln (x^{2} - 25))$

Step 1

Apply the distributive property.

$\frac{1}{3} \cdot 5 + \frac{1}{3} (- \ln (x)) + \frac{1}{3} (- \ln (x^{2} - 25))$

Step 2

Simplify.

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Step 2.1

Combine $\frac{1}{3}$ and $5$ .

$\frac{5}{3} + \frac{1}{3} (- \ln (x)) + \frac{1}{3} (- \ln (x^{2} - 25))$

Step 2.2

Simplify $\frac{1}{3} \ln (x)$ by moving $\frac{1}{3}$ inside the logarithm.

$\frac{5}{3} - \ln (x^{\frac{1}{3}}) + \frac{1}{3} (- \ln (x^{2} - 25))$

Step 2.3

Simplify $\frac{1}{3} \ln (x^{2} - 25)$ by moving $\frac{1}{3}$ inside the logarithm.

$\frac{5}{3} - \ln (x^{\frac{1}{3}}) - \ln ({(x^{2} - 25)}^{\frac{1}{3}})$

Step 3

To write $- \ln ({(x^{2} - 25)}^{\frac{1}{3}})$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{5}{3} - \ln ({(x^{2} - 25)}^{\frac{1}{3}}) \cdot \frac{3}{3} - \ln (x^{\frac{1}{3}})$

Step 4

Simplify terms.

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Step 4.1

Combine $- \ln ({(x^{2} - 25)}^{\frac{1}{3}})$ and $\frac{3}{3}$ .

$\frac{5}{3} + \frac{- \ln ({(x^{2} - 25)}^{\frac{1}{3}}) \cdot 3}{3} - \ln (x^{\frac{1}{3}})$

Step 4.2

Combine the numerators over the common denominator.

$\frac{5 - \ln ({(x^{2} - 25)}^{\frac{1}{3}}) \cdot 3}{3} - \ln (x^{\frac{1}{3}})$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $- \ln ({(x^{2} - 25)}^{\frac{1}{3}}) \cdot 3$ .

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Step 5.1.1

Multiply $3$ by $- 1$ .

$\frac{5 - 3 \ln ({(x^{2} - 25)}^{\frac{1}{3}})}{3} - \ln (x^{\frac{1}{3}})$

Step 5.1.2

Simplify $- 3 \ln ({(x^{2} - 25)}^{\frac{1}{3}})$ by moving $3$ inside the logarithm.

$\frac{5 - \ln ({({(x^{2} - 25)}^{\frac{1}{3}})}^{3})}{3} - \ln (x^{\frac{1}{3}})$

Step 5.2

Multiply the exponents in ${({(x^{2} - 25)}^{\frac{1}{3}})}^{3}$ .

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Step 5.2.1

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

$\frac{5 - \ln ({(x^{2} - 25)}^{\frac{1}{3} \cdot 3})}{3} - \ln (x^{\frac{1}{3}})$

Step 5.2.2

Cancel the common factor of $3$ .

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Step 5.2.2.1

Cancel the common factor.

$\frac{5 - \ln ({(x^{2} - 25)}^{\frac{1}{3} \cdot 3})}{3} - \ln (x^{\frac{1}{3}})$

Step 5.2.2.2

Rewrite the expression.

$\frac{5 - \ln ({(x^{2} - 25)}^{1})}{3} - \ln (x^{\frac{1}{3}})$

Step 5.3

Simplify.

$\frac{5 - \ln (x^{2} - 25)}{3} - \ln (x^{\frac{1}{3}})$