Precalculus Examples

$\frac{1}{4} \cdot \log (a) - 3 \log (b) + 7 \log (c) + 9 \log (b)$

Step 1

Simplify each term.

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Step 1.1

Simplify $\frac{1}{4} \log (a)$ by moving $\frac{1}{4}$ inside the logarithm.

$\log (a^{\frac{1}{4}}) - 3 \log (b) + 7 \log (c) + 9 \log (b)$

Step 1.2

Simplify $- 3 \log (b)$ by moving $3$ inside the logarithm.

$\log (a^{\frac{1}{4}}) - \log (b^{3}) + 7 \log (c) + 9 \log (b)$

Step 1.3

Simplify $7 \log (c)$ by moving $7$ inside the logarithm.

$\log (a^{\frac{1}{4}}) - \log (b^{3}) + \log (c^{7}) + 9 \log (b)$

Step 1.4

Simplify $9 \log (b)$ by moving $9$ inside the logarithm.

$\log (a^{\frac{1}{4}}) - \log (b^{3}) + \log (c^{7}) + \log (b^{9})$

Step 2

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

$\log (\frac{a^{\frac{1}{4}}}{b^{3}}) + \log (c^{7}) + \log (b^{9})$

Step 3

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log (\frac{a^{\frac{1}{4}}}{b^{3}} c^{7}) + \log (b^{9})$

Step 4

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log (\frac{a^{\frac{1}{4}}}{b^{3}} c^{7} b^{9})$

Step 5

Combine $\frac{a^{\frac{1}{4}}}{b^{3}}$ and $c^{7}$ .

$\log (\frac{a^{\frac{1}{4}} c^{7}}{b^{3}} b^{9})$

Step 6

Cancel the common factor of $b^{3}$ .

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Step 6.1

Factor $b^{3}$ out of $b^{9}$ .

$\log (\frac{a^{\frac{1}{4}} c^{7}}{b^{3}} (b^{3} b^{6}))$

Step 6.2

Cancel the common factor.

$\log (\frac{a^{\frac{1}{4}} c^{7}}{b^{3}} (b^{3} b^{6}))$

Step 6.3

Rewrite the expression.

$\log (a^{\frac{1}{4}} c^{7} b^{6})$

Step 7

Reorder factors in $\log (a^{\frac{1}{4}} c^{7} b^{6})$ .

$\log (c^{7} b^{6} a^{\frac{1}{4}})$