Precalculus Examples

$3 \log_{c} (m) - \frac{5}{6} \cdot \log_{c} (n) + \frac{1}{4} \cdot \log_{c} (j) - 6 \log_{c} (k)$

Step 1

Simplify each term.

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Step 1.1

Simplify $3 \log_{c} (m)$ by moving $3$ inside the logarithm.

$\log_{c} (m^{3}) - \frac{5}{6} \cdot \log_{c} (n) + \frac{1}{4} \cdot \log_{c} (j) - 6 \log_{c} (k)$

Step 1.2

Multiply $- \frac{5}{6} \log_{c} (n)$ .

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Step 1.2.1

Reorder $\log_{c} (n)$ and $\frac{5}{6}$ .

$\log_{c} (m^{3}) - (\frac{5}{6} \log_{c} (n)) + \frac{1}{4} \cdot \log_{c} (j) - 6 \log_{c} (k)$

Step 1.2.2

Simplify $\frac{5}{6} \log_{c} (n)$ by moving $\frac{5}{6}$ inside the logarithm.

$\log_{c} (m^{3}) - \log_{c} (n^{\frac{5}{6}}) + \frac{1}{4} \cdot \log_{c} (j) - 6 \log_{c} (k)$

Step 1.3

Simplify $\frac{1}{4} \log_{c} (j)$ by moving $\frac{1}{4}$ inside the logarithm.

$\log_{c} (m^{3}) - \log_{c} (n^{\frac{5}{6}}) + \log_{c} (j^{\frac{1}{4}}) - 6 \log_{c} (k)$

Step 1.4

Simplify $- 6 \log_{c} (k)$ by moving $6$ inside the logarithm.

$\log_{c} (m^{3}) - \log_{c} (n^{\frac{5}{6}}) + \log_{c} (j^{\frac{1}{4}}) - \log_{c} (k^{6})$

Step 2

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

$\log_{c} (\frac{m^{3}}{n^{\frac{5}{6}}}) + \log_{c} (j^{\frac{1}{4}}) - \log_{c} (k^{6})$

Step 3

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

$\log_{c} (\frac{m^{3}}{n^{\frac{5}{6}}} j^{\frac{1}{4}}) - \log_{c} (k^{6})$

Step 4

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

$\log_{c} (\frac{\frac{m^{3}}{n^{\frac{5}{6}}} j^{\frac{1}{4}}}{k^{6}})$

Step 5

Combine $\frac{m^{3}}{n^{\frac{5}{6}}}$ and $j^{\frac{1}{4}}$ .

$\log_{c} (\frac{\frac{m^{3} j^{\frac{1}{4}}}{n^{\frac{5}{6}}}}{k^{6}})$

Step 6

Multiply the numerator by the reciprocal of the denominator.

$\log_{c} (\frac{m^{3} j^{\frac{1}{4}}}{n^{\frac{5}{6}}} \cdot \frac{1}{k^{6}})$

Step 7

Combine.

$\log_{c} (\frac{m^{3} j^{\frac{1}{4}} \cdot 1}{n^{\frac{5}{6}} k^{6}})$

Step 8

Simplify the expression.

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Step 8.1

Multiply $m^{3}$ by $1$ .

$\log_{c} (\frac{j^{\frac{1}{4}} \cdot m^{3}}{n^{\frac{5}{6}} k^{6}})$

Step 8.2

Reorder factors in $\log_{c} (\frac{j^{\frac{1}{4}} m^{3}}{n^{\frac{5}{6}} k^{6}})$ .

$\log_{c} (\frac{m^{3} j^{\frac{1}{4}}}{k^{6} n^{\frac{5}{6}}})$