Precalculus Examples

$2 \log_{b} (m) - \frac{6}{5} \cdot \log_{b} (n) + \frac{1}{3} \cdot \log_{b} (j) - 6 \log_{b} (k)$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

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

$\log_{b} (m^{2}) - \frac{6}{5} \cdot \log_{b} (n) + \frac{1}{3} \cdot \log_{b} (j) - 6 \log_{b} (k)$

Step 1.2

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

Tap for more steps...

Step 1.2.1

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

$\log_{b} (m^{2}) - (\frac{6}{5} \log_{b} (n)) + \frac{1}{3} \cdot \log_{b} (j) - 6 \log_{b} (k)$

Step 1.2.2

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

$\log_{b} (m^{2}) - \log_{b} (n^{\frac{6}{5}}) + \frac{1}{3} \cdot \log_{b} (j) - 6 \log_{b} (k)$

Step 1.3

Simplify $\frac{1}{3} \log_{b} (j)$ by moving $\frac{1}{3}$ inside the logarithm.

$\log_{b} (m^{2}) - \log_{b} (n^{\frac{6}{5}}) + \log_{b} (j^{\frac{1}{3}}) - 6 \log_{b} (k)$

Step 1.4

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

$\log_{b} (m^{2}) - \log_{b} (n^{\frac{6}{5}}) + \log_{b} (j^{\frac{1}{3}}) - \log_{b} (k^{6})$

Step 2

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

$\log_{b} (\frac{m^{2}}{n^{\frac{6}{5}}}) + \log_{b} (j^{\frac{1}{3}}) - \log_{b} (k^{6})$

Step 3

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

$\log_{b} (\frac{m^{2}}{n^{\frac{6}{5}}} j^{\frac{1}{3}}) - \log_{b} (k^{6})$

Step 4

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

$\log_{b} (\frac{\frac{m^{2}}{n^{\frac{6}{5}}} j^{\frac{1}{3}}}{k^{6}})$

Step 5

Combine $\frac{m^{2}}{n^{\frac{6}{5}}}$ and $j^{\frac{1}{3}}$ .

$\log_{b} (\frac{\frac{m^{2} j^{\frac{1}{3}}}{n^{\frac{6}{5}}}}{k^{6}})$

Step 6

Multiply the numerator by the reciprocal of the denominator.

$\log_{b} (\frac{m^{2} j^{\frac{1}{3}}}{n^{\frac{6}{5}}} \cdot \frac{1}{k^{6}})$

Step 7

Combine.

$\log_{b} (\frac{m^{2} j^{\frac{1}{3}} \cdot 1}{n^{\frac{6}{5}} k^{6}})$

Step 8

Simplify the expression.

Tap for more steps...

Step 8.1

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

$\log_{b} (\frac{j^{\frac{1}{3}} \cdot m^{2}}{n^{\frac{6}{5}} k^{6}})$

Step 8.2

Reorder factors in $\log_{b} (\frac{j^{\frac{1}{3}} m^{2}}{n^{\frac{6}{5}} k^{6}})$ .

$\log_{b} (\frac{m^{2} j^{\frac{1}{3}}}{k^{6} n^{\frac{6}{5}}})$