Expand the Logarithmic Expression log of 4x^3y square root of z

$\log (4 x^{3} y \sqrt{z})$

Step 1

Rewrite $\log (4 x^{3} y \sqrt{z})$ as $\log (4 x^{3} y) + \log (\sqrt{z})$ .

$\log (4 x^{3} y) + \log (\sqrt{z})$

Step 2

Rewrite $\log (4 x^{3} y)$ as $\log (4 x^{3}) + \log (y)$ .

$\log (4 x^{3}) + \log (y) + \log (\sqrt{z})$

Step 3

Rewrite $\log (4 x^{3})$ as $\log (4) + \log (x^{3})$ .

$\log (4) + \log (x^{3}) + \log (y) + \log (\sqrt{z})$

Step 4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{z}$ as $z^{\frac{1}{2}}$ .

$\log (4) + \log (x^{3}) + \log (y) + \log (z^{\frac{1}{2}})$

Step 5

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

$\log (4) + 3 \log (x) + \log (y) + \log (z^{\frac{1}{2}})$

Step 6

Expand $\log (z^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\log (4) + 3 \log (x) + \log (y) + \frac{1}{2} \log (z)$

Step 7

Simplify each term.

Rewrite $\log (4)$ as $\log (2^{2})$ .

$\log (2^{2}) + 3 \log (x) + \log (y) + \frac{1}{2} \log (z)$

Expand $\log (2^{2})$ by moving $2$ outside the logarithm.

$2 \log (2) + 3 \log (x) + \log (y) + \frac{1}{2} \log (z)$

Combine $\frac{1}{2}$ and $\log (z)$ .

$2 \log (2) + 3 \log (x) + \log (y) + \frac{\log (z)}{2}$