Algebra Examples

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

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

Step 1

Simplify each term.

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Step 1.1

Simplify

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

$\frac{1}{3} \ln ({(x + 2)}^{3})$ by moving

\frac{1}{3}

$\frac{1}{3}$ inside the logarithm.

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

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

Step 1.2

Multiply the exponents in

{({(x + 2)}^{3})}^{\frac{1}{3}}

${({(x + 2)}^{3})}^{\frac{1}{3}}$ .

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Step 1.2.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 1.2.2

Cancel the common factor of

3

$3$ .

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Step 1.2.2.1

Cancel the common factor.

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

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

Step 1.2.2.2

Rewrite the expression.

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

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

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

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

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

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

Step 1.3

Simplify.

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

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

Step 1.4

Use the quotient property of logarithms,

{log}_{b} (x) - {log}_{b} (y) = {log}_{b} (\frac{x}{y})

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

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

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

Step 1.5

Simplify the denominator.

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Step 1.5.1

Factor

x^{2} + 3 x + 2

$x^{2} + 3 x + 2$ using the AC method.

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Step 1.5.1.1

Consider the form

x^{2} + b x + c

$x^{2} + b x + c$ . Find a pair of integers whose product is

c

$c$ and whose sum is

b

$b$ . In this case, whose product is

2

$2$ and whose sum is

3

$3$ .

1, 2

$1, 2$

Step 1.5.1.2

Write the factored form using these integers.

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

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

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

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

Step 1.5.2

Apply the product rule to

(x + 1) (x + 2)

$(x + 1) (x + 2)$ .

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

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

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

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

Step 1.6

Simplify

\frac{1}{2} ln (\frac{x}{{(x + 1)}^{2} {(x + 2)}^{2}})

$\frac{1}{2} \ln (\frac{x}{{(x + 1)}^{2} {(x + 2)}^{2}})$ by moving

\frac{1}{2}

$\frac{1}{2}$ inside the logarithm.

ln (x + 2) + ln ⎛ ⎜ ⎝ {(\frac{x}{{(x + 1)}^{2} {(x + 2)}^{2}})}^{\frac{1}{2}} ⎞ ⎟ ⎠

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

Step 1.7

Use the power rule

{(a b)}^{n} = a^{n} b^{n}

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

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Step 1.7.1

Apply the product rule to

\frac{x}{{(x + 1)}^{2} {(x + 2)}^{2}}

$\frac{x}{{(x + 1)}^{2} {(x + 2)}^{2}}$ .

ln (x + 2) + ln ⎛ ⎜ ⎜ ⎜ ⎝ \frac{x^{\frac{1}{2}}}{{({(x + 1)}^{2} {(x + 2)}^{2})}^{\frac{1}{2}}} ⎞ ⎟ ⎟ ⎟ ⎠

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

Step 1.7.2

Apply the product rule to

{(x + 1)}^{2} {(x + 2)}^{2}

${(x + 1)}^{2} {(x + 2)}^{2}$ .

ln (x + 2) + ln ⎛ ⎜ ⎜ ⎜ ⎝ \frac{x^{\frac{1}{2}}}{{({(x + 1)}^{2})}^{\frac{1}{2}} {({(x + 2)}^{2})}^{\frac{1}{2}}} ⎞ ⎟ ⎟ ⎟ ⎠

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

ln (x + 2) + ln ⎛ ⎜ ⎜ ⎜ ⎝ \frac{x^{\frac{1}{2}}}{{({(x + 1)}^{2})}^{\frac{1}{2}} {({(x + 2)}^{2})}^{\frac{1}{2}}} ⎞ ⎟ ⎟ ⎟ ⎠

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

Step 1.8

Simplify the denominator.

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Step 1.8.1

Multiply the exponents in

{({(x + 1)}^{2})}^{\frac{1}{2}}

${({(x + 1)}^{2})}^{\frac{1}{2}}$ .

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Step 1.8.1.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

ln (x + 2) + ln ⎛ ⎜ ⎜ ⎜ ⎝ \frac{x^{\frac{1}{2}}}{{(x + 1)}^{2 (\frac{1}{2})} {({(x + 2)}^{2})}^{\frac{1}{2}}} ⎞ ⎟ ⎟ ⎟ ⎠

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

Step 1.8.1.2

Cancel the common factor of

2

$2$ .

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Step 1.8.1.2.1

Cancel the common factor.

ln (x + 2) + ln ⎛ ⎜ ⎜ ⎜ ⎝ \frac{x^{\frac{1}{2}}}{{(x + 1)}^{2 (\frac{1}{2})} {({(x + 2)}^{2})}^{\frac{1}{2}}} ⎞ ⎟ ⎟ ⎟ ⎠

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

Step 1.8.1.2.2

Rewrite the expression.

ln (x + 2) + ln ⎛ ⎜ ⎜ ⎜ ⎝ \frac{x^{\frac{1}{2}}}{{(x + 1)}^{1} {({(x + 2)}^{2})}^{\frac{1}{2}}} ⎞ ⎟ ⎟ ⎟ ⎠

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

ln (x + 2) + ln ⎛ ⎜ ⎜ ⎜ ⎝ \frac{x^{\frac{1}{2}}}{{(x + 1)}^{1} {({(x + 2)}^{2})}^{\frac{1}{2}}} ⎞ ⎟ ⎟ ⎟ ⎠

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

ln (x + 2) + ln ⎛ ⎜ ⎜ ⎜ ⎝ \frac{x^{\frac{1}{2}}}{{(x + 1)}^{1} {({(x + 2)}^{2})}^{\frac{1}{2}}} ⎞ ⎟ ⎟ ⎟ ⎠

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

Step 1.8.2

Simplify.

ln (x + 2) + ln ⎛ ⎜ ⎜ ⎜ ⎝ \frac{x^{\frac{1}{2}}}{(x + 1) {({(x + 2)}^{2})}^{\frac{1}{2}}} ⎞ ⎟ ⎟ ⎟ ⎠

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

Step 1.8.3

Multiply the exponents in

{({(x + 2)}^{2})}^{\frac{1}{2}}

${({(x + 2)}^{2})}^{\frac{1}{2}}$ .

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Step 1.8.3.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

ln (x + 2) + ln ⎛ ⎜ ⎝ \frac{x^{\frac{1}{2}}}{(x + 1) {(x + 2)}^{2 (\frac{1}{2})}} ⎞ ⎟ ⎠

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

Step 1.8.3.2

Cancel the common factor of

2

$2$ .

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Step 1.8.3.2.1

Cancel the common factor.

ln (x + 2) + ln ⎛ ⎜ ⎝ \frac{x^{\frac{1}{2}}}{(x + 1) {(x + 2)}^{2 (\frac{1}{2})}} ⎞ ⎟ ⎠

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

Step 1.8.3.2.2

Rewrite the expression.

ln (x + 2) + ln (\frac{x^{\frac{1}{2}}}{(x + 1) {(x + 2)}^{1}})

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

Step 1.8.4

Simplify.

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

Step 2

Use the product property of logarithms,

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

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

Step 3

Cancel the common factor of

$x + 2$ .

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Step 3.1

Factor

$x + 2$ out of

$(x + 1) (x + 2)$ .

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

Step 3.2

Cancel the common factor.

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

Step 3.3

Rewrite the expression.

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