Precalculus Examples

$\log (a) - 6 (21 \log (b) - \log (c))$

Step 1

Simplify each term.

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Step 1.1

Simplify $21 \log (b)$ by moving $21$ inside the logarithm.

$\log (a) - 6 (\log (b^{21}) - \log (c))$

Step 1.2

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

$\log (a) - 6 \log (\frac{b^{21}}{c})$

Step 1.3

Simplify $- 6 \log (\frac{b^{21}}{c})$ by moving $6$ inside the logarithm.

$\log (a) - \log ({(\frac{b^{21}}{c})}^{6})$

Step 1.4

Apply the product rule to $\frac{b^{21}}{c}$ .

$\log (a) - \log (\frac{{(b^{21})}^{6}}{c^{6}})$

Step 1.5

Multiply the exponents in ${(b^{21})}^{6}$ .

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Step 1.5.1

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

$\log (a) - \log (\frac{b^{21 \cdot 6}}{c^{6}})$

Step 1.5.2

Multiply $21$ by $6$ .

$\log (a) - \log (\frac{b^{126}}{c^{6}})$

Step 2

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

$\log (\frac{a}{\frac{b^{126}}{c^{6}}})$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\log (a \frac{c^{6}}{b^{126}})$

Step 4

Combine $a$ and $\frac{c^{6}}{b^{126}}$ .

$\log (\frac{a c^{6}}{b^{126}})$