Calculus Examples

ln (5) + \frac{1}{2} \cdot ln (x + 3) - 3 ln (1 + \sqrt{x})

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

Step 1

Simplify each term.

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Step 1.1

Simplify

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

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

\frac{1}{2}

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

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

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

Step 1.2

Simplify

- 3 ln (1 + \sqrt{x})

$- 3 \ln (1 + \sqrt{x})$ by moving

3

$3$ inside the logarithm.

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

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

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

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

Step 2

Use the product property of logarithms,

{log}_{b} (x) + {log}_{b} (y) = {log}_{b} (x y)

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

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

Step 3

Use the quotient property of logarithms,

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

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

Step 4

Use the Binomial Theorem.

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

Step 5

Simplify each term.

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Step 5.1

One to any power is one.

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

Step 5.2

One to any power is one.

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

Step 5.3

Multiply

$3$ by

$1$ .

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

Step 5.4

Multiply

$3$ by

$1$ .

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

Step 5.5

Rewrite

${\sqrt{x}}^{2}$ as

$x$ .

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Step 5.5.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{x}$ as

$x^{\frac{1}{2}}$ .

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

Step 5.5.2

Apply the power rule and multiply exponents,

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

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

Step 5.5.3

Combine

$\frac{1}{2}$ and

$2$ .

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

Step 5.5.4

Cancel the common factor of

$2$ .

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Step 5.5.4.1

Cancel the common factor.

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

Step 5.5.4.2

Rewrite the expression.

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

Step 5.5.5

Simplify.

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

Step 5.6

Rewrite

${\sqrt{x}}^{3}$ as

$\sqrt{x^{3}}$ .

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

Step 5.7

Factor out

$x^{2}$ .

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

Step 5.8

Pull terms out from under the radical.

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

Step 6

Simplify the denominator.

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Step 6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{x}$ as

$x^{\frac{1}{2}}$ .

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

Step 6.2

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{x}$ as

$x^{\frac{1}{2}}$ .

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

Step 6.3

Multiply

$x$ by

$x^{\frac{1}{2}}$ by adding the exponents.

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Step 6.3.1

Multiply

$x$ by

$x^{\frac{1}{2}}$ .

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Step 6.3.1.1

Raise

$x$ to the power of

$1$ .

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

Step 6.3.1.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 6.3.2

Write

$1$ as a fraction with a common denominator.

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

Step 6.3.3

Combine the numerators over the common denominator.

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

Step 6.3.4

Add

$2$ and

$1$ .

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

Step 6.4

Reorder terms.

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

Step 6.5

Rewrite

$3 x + 3 x^{\frac{1}{2}} + x^{\frac{3}{2}} + 1$ in a factored form.

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Step 6.5.1

Factor

$3 x^{\frac{1}{2}}$ out of

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

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Step 6.5.1.1

Factor

$3 x^{\frac{1}{2}}$ out of

$3 x$ .

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

Step 6.5.1.2

Factor

$3 x^{\frac{1}{2}}$ out of

$3 x^{\frac{1}{2}}$ .

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

Step 6.5.1.3

Factor

$3 x^{\frac{1}{2}}$ out of

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

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

Step 6.5.2

Rewrite

$x^{\frac{3}{2}}$ as

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

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

Step 6.5.3

Rewrite

$1$ as

$1^{3}$ .

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

Step 6.5.4

Since both terms are perfect cubes, factor using the sum of cubes formula,

$a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where

$a = x^{\frac{1}{2}}$ and

$b = 1$ .

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

Step 6.5.5

Simplify.

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Step 6.5.5.1

Multiply the exponents in

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

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Step 6.5.5.1.1

Apply the power rule and multiply exponents,

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

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

Step 6.5.5.1.2

Cancel the common factor of

$2$ .

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Step 6.5.5.1.2.1

Cancel the common factor.

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

Step 6.5.5.1.2.2

Rewrite the expression.

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

Step 6.5.5.2

Simplify.

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

Step 6.5.5.3

Multiply

$- 1$ by

$1$ .

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

Step 6.5.5.4

One to any power is one.

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

Step 6.5.6

Factor

$x^{\frac{1}{2}} + 1$ out of

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

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Step 6.5.6.1

Factor

$x^{\frac{1}{2}} + 1$ out of

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

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

Step 6.5.6.2

Factor

$x^{\frac{1}{2}} + 1$ out of

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

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

Step 6.5.7

Subtract

$x^{\frac{1}{2}}$ from

$3 x^{\frac{1}{2}}$ .

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

Step 6.5.8

Rewrite

$x$ as

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

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

Step 6.5.9

Let

$u = x^{\frac{1}{2}}$ . Substitute

$u$ for all occurrences of

$x^{\frac{1}{2}}$ .

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

Step 6.5.10

Factor using the perfect square rule.

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Step 6.5.10.1

Rewrite

$1$ as

$1^{2}$ .

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

Step 6.5.10.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 u = 2 \cdot u \cdot 1$

Step 6.5.10.3

Rewrite the polynomial.

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

Step 6.5.10.4

Factor using the perfect square trinomial rule

$a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where

$a = u$ and

$b = 1$ .

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

Step 6.5.11

Replace all occurrences of

$u$ with

$x^{\frac{1}{2}}$ .

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

Step 7

Multiply

$x^{\frac{1}{2}} + 1$ by

${(x^{\frac{1}{2}} + 1)}^{2}$ by adding the exponents.

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Step 7.1

Multiply

$x^{\frac{1}{2}} + 1$ by

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

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Step 7.1.1

Raise

$x^{\frac{1}{2}} + 1$ to the power of

$1$ .

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

Step 7.1.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 7.2

Add

$1$ and

$2$ .

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