Write as a Single Logarithm log of x-2( log of y+3 log of z)

$\log (x) - 2 (\log (y) + 3 \log (z))$

Step 1

Simplify each term.

Simplify $3 \log (z)$ by moving $3$ inside the logarithm.

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

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

$\log (x) - 2 \log (y z^{3})$

Simplify $- 2 \log (y z^{3})$ by moving $2$ inside the logarithm.

$\log (x) - \log ({(y z^{3})}^{2})$

Apply the product rule to $y z^{3}$ .

$\log (x) - \log (y^{2} {(z^{3})}^{2})$

Multiply the exponents in ${(z^{3})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log (x) - \log (y^{2} z^{3 \cdot 2})$

Multiply $3$ by $2$ .

$\log (x) - \log (y^{2} z^{6})$

Step 2

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

$\log (\frac{x}{y^{2} z^{6}})$