Precalculus Examples

$\log_{b} (\frac{b \sqrt{b}}{b^{2} {(c + 1)}^{3} (d - 2)})$

Step 1

Rewrite $\log_{b} (\frac{b \sqrt{b}}{b^{2} {(c + 1)}^{3} (d - 2)})$ as $\log_{b} (b \sqrt{b}) - \log_{b} (b^{2} {(c + 1)}^{3} (d - 2))$ .

$\log_{b} (b \sqrt{b}) - \log_{b} (b^{2} {(c + 1)}^{3} (d - 2))$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{b}$ as $b^{\frac{1}{2}}$ .

$\log_{b} (b \cdot b^{\frac{1}{2}}) - \log_{b} (b^{2} {(c + 1)}^{3} (d - 2))$

Step 3

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

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Step 3.1

Multiply $b$ by $b^{\frac{1}{2}}$ .

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Step 3.1.1

Raise $b$ to the power of $1$ .

$\log_{b} (b^{1} b^{\frac{1}{2}}) - \log_{b} (b^{2} {(c + 1)}^{3} (d - 2))$

Step 3.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\log_{b} (b^{1 + \frac{1}{2}}) - \log_{b} (b^{2} {(c + 1)}^{3} (d - 2))$

Step 3.2

Write $1$ as a fraction with a common denominator.

$\log_{b} (b^{\frac{2}{2} + \frac{1}{2}}) - \log_{b} (b^{2} {(c + 1)}^{3} (d - 2))$

Step 3.3

Combine the numerators over the common denominator.

$\log_{b} (b^{\frac{2 + 1}{2}}) - \log_{b} (b^{2} {(c + 1)}^{3} (d - 2))$

Step 3.4

Add $2$ and $1$ .

$\log_{b} (b^{\frac{3}{2}}) - \log_{b} (b^{2} {(c + 1)}^{3} (d - 2))$

Step 4

Expand $\log_{b} (b^{\frac{3}{2}})$ by moving $\frac{3}{2}$ outside the logarithm.

$\frac{3}{2} \log_{b} (b) - \log_{b} (b^{2} {(c + 1)}^{3} (d - 2))$

Step 5

Logarithm base $b$ of $b$ is $1$ .

$\frac{3}{2} \cdot 1 - \log_{b} (b^{2} {(c + 1)}^{3} (d - 2))$

Step 6

Multiply $\frac{3}{2}$ by $1$ .

$\frac{3}{2} - \log_{b} (b^{2} {(c + 1)}^{3} (d - 2))$

Step 7

Rewrite $\log_{b} (b^{2} {(c + 1)}^{3} (d - 2))$ as $\log_{b} (b^{2} {(c + 1)}^{3}) + \log_{b} (d - 2)$ .

$\frac{3}{2} - (\log_{b} (b^{2} {(c + 1)}^{3}) + \log_{b} (d - 2))$

Step 8

Rewrite $\log_{b} (b^{2} {(c + 1)}^{3})$ as $\log_{b} (b^{2}) + \log_{b} ({(c + 1)}^{3})$ .

$\frac{3}{2} - (\log_{b} (b^{2}) + \log_{b} ({(c + 1)}^{3}) + \log_{b} (d - 2))$

Step 9

Expand $\log_{b} (b^{2})$ by moving $2$ outside the logarithm.

$\frac{3}{2} - (2 \log_{b} (b) + \log_{b} ({(c + 1)}^{3}) + \log_{b} (d - 2))$

Step 10

Expand $\log_{b} ({(c + 1)}^{3})$ by moving $3$ outside the logarithm.

$\frac{3}{2} - (2 \log_{b} (b) + 3 \log_{b} (c + 1) + \log_{b} (d - 2))$

Step 11

Logarithm base $b$ of $b$ is $1$ .

$\frac{3}{2} - (2 \cdot 1 + 3 \log_{b} (c + 1) + \log_{b} (d - 2))$

Step 12

Multiply $2$ by $1$ .

$\frac{3}{2} - (2 + 3 \log_{b} (c + 1) + \log_{b} (d - 2))$

Step 13

Simplify each term.

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Step 13.1

Apply the distributive property.

$\frac{3}{2} - 1 \cdot 2 - (3 \log_{b} (c + 1)) - \log_{b} (d - 2)$

Step 13.2

Simplify.

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Step 13.2.1

Multiply $- 1$ by $2$ .

$\frac{3}{2} - 2 - (3 \log_{b} (c + 1)) - \log_{b} (d - 2)$

Step 13.2.2

Multiply $3$ by $- 1$ .

$\frac{3}{2} - 2 - 3 \log_{b} (c + 1) - \log_{b} (d - 2)$

Step 14

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3}{2} - 2 \cdot \frac{2}{2} - 3 \log_{b} (c + 1) - \log_{b} (d - 2)$

Step 15

Combine $- 2$ and $\frac{2}{2}$ .

$\frac{3}{2} + \frac{- 2 \cdot 2}{2} - 3 \log_{b} (c + 1) - \log_{b} (d - 2)$

Step 16

Combine the numerators over the common denominator.

$\frac{3 - 2 \cdot 2}{2} - 3 \log_{b} (c + 1) - \log_{b} (d - 2)$

Step 17

Simplify the numerator.

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Step 17.1

Multiply $- 2$ by $2$ .

$\frac{3 - 4}{2} - 3 \log_{b} (c + 1) - \log_{b} (d - 2)$

Step 17.2

Subtract $4$ from $3$ .

$\frac{- 1}{2} - 3 \log_{b} (c + 1) - \log_{b} (d - 2)$

Step 18

Move the negative in front of the fraction.

$- \frac{1}{2} - 3 \log_{b} (c + 1) - \log_{b} (d - 2)$

Step 19

To write $- 3 \log_{b} (c + 1)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{1}{2} - 3 \log_{b} (c + 1) \cdot \frac{2}{2} - \log_{b} (d - 2)$

Step 20

Combine $- 3 \log_{b} (c + 1)$ and $\frac{2}{2}$ .

$- \frac{1}{2} + \frac{- 3 \log_{b} (c + 1) \cdot 2}{2} - \log_{b} (d - 2)$

Step 21

Combine the numerators over the common denominator.

$\frac{- 1 - 3 \log_{b} (c + 1) \cdot 2}{2} - \log_{b} (d - 2)$

Step 22

Simplify each term.

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Step 22.1

Simplify the numerator.

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Step 22.1.1

Factor $- 1$ out of $- 1 - 3 \log_{b} (c + 1) \cdot 2$ .

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Step 22.1.1.1

Move $\log_{b} (c + 1)$ .

$\frac{- 1 - 3 \cdot 2 \log_{b} (c + 1)}{2} - \log_{b} (d - 2)$

Step 22.1.1.2

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{- 1 (1) - 3 \cdot 2 \log_{b} (c + 1)}{2} - \log_{b} (d - 2)$

Step 22.1.1.3

Factor $- 1$ out of $- 3 \cdot 2 \log_{b} (c + 1)$ .

$\frac{- 1 (1) - (3 \cdot 2 \log_{b} (c + 1))}{2} - \log_{b} (d - 2)$

Step 22.1.1.4

Factor $- 1$ out of $- 1 (1) - (3 \cdot 2 \log_{b} (c + 1))$ .

$\frac{- 1 (1 + 3 \cdot 2 \log_{b} (c + 1))}{2} - \log_{b} (d - 2)$

Step 22.1.1.5

Rewrite $- 1 (1 + 3 \cdot 2 \log_{b} (c + 1))$ as $- (1 + 3 \cdot 2 \log_{b} (c + 1))$ .

$\frac{- (1 + 3 \cdot 2 \log_{b} (c + 1))}{2} - \log_{b} (d - 2)$

Step 22.1.2

Multiply $3$ by $2$ .

$\frac{- (1 + 6 \log_{b} (c + 1))}{2} - \log_{b} (d - 2)$

Step 22.2

Move the negative in front of the fraction.

$- \frac{1 + 6 \log_{b} (c + 1)}{2} - \log_{b} (d - 2)$

Step 23

To write $- \log_{b} (d - 2)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{1 + 6 \log_{b} (c + 1)}{2} - \log_{b} (d - 2) \cdot \frac{2}{2}$

Step 24

Combine $- \log_{b} (d - 2)$ and $\frac{2}{2}$ .

$- \frac{1 + 6 \log_{b} (c + 1)}{2} + \frac{- \log_{b} (d - 2) \cdot 2}{2}$

Step 25

Combine the numerators over the common denominator.

$\frac{- (1 + 6 \log_{b} (c + 1)) - \log_{b} (d - 2) \cdot 2}{2}$

Step 26

Factor $- 1$ out of $- (1 + 6 \log_{b} (c + 1)) - \log_{b} (d - 2) \cdot 2$ .

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Step 26.1

Move $\log_{b} (d - 2)$ .

$\frac{- (1 + 6 \log_{b} (c + 1)) - 1 \cdot 2 \log_{b} (d - 2)}{2}$

Step 26.2

Factor $- 1$ out of $- 1 \cdot 2 \log_{b} (d - 2)$ .

$\frac{- (1 + 6 \log_{b} (c + 1)) - (2 \log_{b} (d - 2))}{2}$

Step 26.3

Factor $- 1$ out of $- (1 + 6 \log_{b} (c + 1)) - (2 \log_{b} (d - 2))$ .

$\frac{- (1 + 6 \log_{b} (c + 1) + 2 \log_{b} (d - 2))}{2}$

Step 27

Move the negative in front of the fraction.

$- \frac{1 + 6 \log_{b} (c + 1) + 2 \log_{b} (d - 2)}{2}$