Precalculus Examples

$\log_{3} (\frac{27 x^{3} y^{\frac{3}{2}}}{64})$

Step 1

Rewrite $\log_{3} (\frac{27 x^{3} y^{\frac{3}{2}}}{64})$ as $\log_{3} (27 x^{3} y^{\frac{3}{2}}) - \log_{3} (64)$ .

$\log_{3} (27 x^{3} y^{\frac{3}{2}}) - \log_{3} (64)$

Step 2

Rewrite $\log_{3} (64)$ as $\log_{3} (2^{6})$ .

$\log_{3} (27 x^{3} y^{\frac{3}{2}}) - \log_{3} (2^{6})$

Step 3

Expand $\log_{3} (2^{6})$ by moving $6$ outside the logarithm.

$\log_{3} (27 x^{3} y^{\frac{3}{2}}) - (6 \log_{3} (2))$

Step 4

Multiply $6$ by $- 1$ .

$\log_{3} (27 x^{3} y^{\frac{3}{2}}) - 6 \log_{3} (2)$

Step 5

Rewrite $\log_{3} (27 x^{3} y^{\frac{3}{2}})$ as $\log_{3} (27 x^{3}) + \log_{3} (y^{\frac{3}{2}})$ .

$\log_{3} (27 x^{3}) + \log_{3} (y^{\frac{3}{2}}) - 6 \log_{3} (2)$

Step 6

Rewrite $\log_{3} (27 x^{3})$ as $\log_{3} (27) + \log_{3} (x^{3})$ .

$\log_{3} (27) + \log_{3} (x^{3}) + \log_{3} (y^{\frac{3}{2}}) - 6 \log_{3} (2)$

Step 7

Expand $\log_{3} (x^{3})$ by moving $3$ outside the logarithm.

$\log_{3} (27) + 3 \log_{3} (x) + \log_{3} (y^{\frac{3}{2}}) - 6 \log_{3} (2)$

Step 8

Expand $\log_{3} (y^{\frac{3}{2}})$ by moving $\frac{3}{2}$ outside the logarithm.

$\log_{3} (27) + 3 \log_{3} (x) + \frac{3}{2} \log_{3} (y) - 6 \log_{3} (2)$

Step 9

Logarithm base $3$ of $27$ is $3$ .

$3 + 3 \log_{3} (x) + \frac{3}{2} \log_{3} (y) - 6 \log_{3} (2)$

Step 10

Combine $\frac{3}{2}$ and $\log_{3} (y)$ .

$3 + 3 \log_{3} (x) + \frac{3 \log_{3} (y)}{2} - 6 \log_{3} (2)$