Precalculus Examples

$2 \log_{c} (m) - \frac{5}{3} \cdot \log_{c} (n) + \frac{1}{4} \cdot \log_{c} (j) - 5 \log_{c} (k)$

Step 1

Simplify each term.

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Step 1.1

Simplify $2 \log_{c} (m)$ by moving $2$ inside the logarithm.

$\log_{c} (m^{2}) - \frac{5}{3} \cdot \log_{c} (n) + \frac{1}{4} \cdot \log_{c} (j) - 5 \log_{c} (k)$

Step 1.2

Multiply $- \frac{5}{3} \log_{c} (n)$ .

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Step 1.2.1

Reorder $\log_{c} (n)$ and $\frac{5}{3}$ .

$\log_{c} (m^{2}) - (\frac{5}{3} \log_{c} (n)) + \frac{1}{4} \cdot \log_{c} (j) - 5 \log_{c} (k)$

Step 1.2.2

Simplify $\frac{5}{3} \log_{c} (n)$ by moving $\frac{5}{3}$ inside the logarithm.

$\log_{c} (m^{2}) - \log_{c} (n^{\frac{5}{3}}) + \frac{1}{4} \cdot \log_{c} (j) - 5 \log_{c} (k)$

Step 1.3

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

$\log_{c} (m^{2}) - \log_{c} (n^{\frac{5}{3}}) + \log_{c} (j^{\frac{1}{4}}) - 5 \log_{c} (k)$

Step 1.4

Simplify $- 5 \log_{c} (k)$ by moving $5$ inside the logarithm.

$\log_{c} (m^{2}) - \log_{c} (n^{\frac{5}{3}}) + \log_{c} (j^{\frac{1}{4}}) - \log_{c} (k^{5})$

Step 2

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

$\log_{c} (\frac{m^{2}}{n^{\frac{5}{3}}}) + \log_{c} (j^{\frac{1}{4}}) - \log_{c} (k^{5})$

Step 3

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

$\log_{c} (\frac{m^{2}}{n^{\frac{5}{3}}} j^{\frac{1}{4}}) - \log_{c} (k^{5})$

Step 4

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

$\log_{c} (\frac{\frac{m^{2}}{n^{\frac{5}{3}}} j^{\frac{1}{4}}}{k^{5}})$

Step 5

Combine $\frac{m^{2}}{n^{\frac{5}{3}}}$ and $j^{\frac{1}{4}}$ .

$\log_{c} (\frac{\frac{m^{2} j^{\frac{1}{4}}}{n^{\frac{5}{3}}}}{k^{5}})$

Step 6

Multiply the numerator by the reciprocal of the denominator.

$\log_{c} (\frac{m^{2} j^{\frac{1}{4}}}{n^{\frac{5}{3}}} \cdot \frac{1}{k^{5}})$

Step 7

Combine.

$\log_{c} (\frac{m^{2} j^{\frac{1}{4}} \cdot 1}{n^{\frac{5}{3}} k^{5}})$

Step 8

Simplify the expression.

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Step 8.1

Multiply $m^{2}$ by $1$ .

$\log_{c} (\frac{j^{\frac{1}{4}} \cdot m^{2}}{n^{\frac{5}{3}} k^{5}})$

Step 8.2

Reorder factors in $\log_{c} (\frac{j^{\frac{1}{4}} m^{2}}{n^{\frac{5}{3}} k^{5}})$ .

$\log_{c} (\frac{m^{2} j^{\frac{1}{4}}}{k^{5} n^{\frac{5}{3}}})$