Simplify/Condense log base w of 5-2 log base w of a+3 log base w of b-4 log base w of c

$\log_{w} (5) - 2 \log_{w} (a) + 3 \log_{w} (b) - 4 \log_{w} (c)$

Simplify each term.

Simplify $- 2 \log_{w} (a)$ by moving $2$ inside the logarithm.

$\log_{w} (5) - \log_{w} (a^{2}) + 3 \log_{w} (b) - 4 \log_{w} (c)$

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

$\log_{w} (5) - \log_{w} (a^{2}) + \log_{w} (b^{3}) - 4 \log_{w} (c)$

Simplify $- 4 \log_{w} (c)$ by moving $4$ inside the logarithm.

$\log_{w} (5) - \log_{w} (a^{2}) + \log_{w} (b^{3}) - \log_{w} (c^{4})$

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

$\log_{w} (\frac{5}{a^{2}}) + \log_{w} (b^{3}) - \log_{w} (c^{4})$

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

$\log_{w} (\frac{5}{a^{2}} b^{3}) - \log_{w} (c^{4})$

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

$\log_{w} (\frac{\frac{5}{a^{2}} b^{3}}{c^{4}})$

Combine $\frac{5}{a^{2}}$ and $b^{3}$ .

$\log_{w} (\frac{\frac{5 b^{3}}{a^{2}}}{c^{4}})$

Multiply the numerator by the reciprocal of the denominator.

$\log_{w} (\frac{5 b^{3}}{a^{2}} \cdot \frac{1}{c^{4}})$

Combine.

$\log_{w} (\frac{5 b^{3} \cdot 1}{a^{2} c^{4}})$

Multiply $5$ by $1$ .

$\log_{w} (\frac{5 b^{3}}{a^{2} c^{4}})$