Precalculus Examples

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

Step 1

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

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

Step 2

Simplify each term.

Tap for more steps...

Step 2.1

Apply the distributive property.

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

Step 2.2

Multiply $3$ by $2$ .

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

Step 2.3

Simplify the denominator.

Tap for more steps...

Step 2.3.1

Rewrite $1$ as $1^{2}$ .

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

Step 2.3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

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

Step 3

Apply the distributive property.

$\frac{1}{3} (2 h x) + \frac{1}{3} (6 h) + \frac{1}{3} \ln (\frac{x}{(x + 1) (x - 1)})$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Multiply $\frac{1}{3} (2 h x)$ .

Tap for more steps...

Step 4.1.1

Combine $2$ and $\frac{1}{3}$ .

$\frac{2}{3} (h x) + \frac{1}{3} (6 h) + \frac{1}{3} \ln (\frac{x}{(x + 1) (x - 1)})$

Step 4.1.2

Combine $h$ and $\frac{2}{3}$ .

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

Step 4.1.3

Combine $\frac{h \cdot 2}{3}$ and $x$ .

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

Step 4.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.2.1

Factor $3$ out of $6 h$ .

$\frac{h \cdot 2 x}{3} + \frac{1}{3} (3 (2 h)) + \frac{1}{3} \ln (\frac{x}{(x + 1) (x - 1)})$

Step 4.2.2

Cancel the common factor.

$\frac{h \cdot 2 x}{3} + \frac{1}{3} (3 (2 h)) + \frac{1}{3} \ln (\frac{x}{(x + 1) (x - 1)})$

Step 4.2.3

Rewrite the expression.

$\frac{h \cdot 2 x}{3} + 2 h + \frac{1}{3} \ln (\frac{x}{(x + 1) (x - 1)})$

Step 4.3

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

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

Step 5

Simplify each term.

Tap for more steps...

Step 5.1

Move $2$ to the left of $h$ .

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

Step 5.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 5.2.1

Apply the product rule to $\frac{x}{(x + 1) (x - 1)}$ .

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

Step 5.2.2

Apply the product rule to $(x + 1) (x - 1)$ .

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

Step 6

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

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

Step 7

Simplify terms.

Tap for more steps...

Step 7.1

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

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

Step 7.2

Combine the numerators over the common denominator.

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

Step 8

Simplify the numerator.

Tap for more steps...

Step 8.1

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

Tap for more steps...

Step 8.1.1

Reorder $\ln (\frac{x^{\frac{1}{3}}}{{(x + 1)}^{\frac{1}{3}} {(x - 1)}^{\frac{1}{3}}})$ and $3$ .

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

Step 8.1.2

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

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

Step 8.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 8.2.1

Apply the product rule to $\frac{x^{\frac{1}{3}}}{{(x + 1)}^{\frac{1}{3}} {(x - 1)}^{\frac{1}{3}}}$ .

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

Step 8.2.2

Apply the product rule to ${(x + 1)}^{\frac{1}{3}} {(x - 1)}^{\frac{1}{3}}$ .

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

Step 8.3

Simplify the numerator.

Tap for more steps...

Step 8.3.1

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

Tap for more steps...

Step 8.3.1.1

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

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

Step 8.3.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 8.3.1.2.1

Cancel the common factor.

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

Step 8.3.1.2.2

Rewrite the expression.

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

Step 8.3.2

Simplify.

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

Step 8.4

Simplify the denominator.

Tap for more steps...

Step 8.4.1

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

Tap for more steps...

Step 8.4.1.1

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

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

Step 8.4.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 8.4.1.2.1

Cancel the common factor.

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

Step 8.4.1.2.2

Rewrite the expression.

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

Step 8.4.2

Simplify.

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

Step 8.4.3

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

Tap for more steps...

Step 8.4.3.1

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

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

Step 8.4.3.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 8.4.3.2.1

Cancel the common factor.

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

Step 8.4.3.2.2

Rewrite the expression.

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

Step 8.4.4

Simplify.

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