Write as a Single Logarithm 3 log base 5 of (uv^2)/(w^3)

$3 \log_{5} (\frac{u v^{2}}{w^{3}})$

Step 1

Simplify $3 \log_{5} (\frac{u v^{2}}{w^{3}})$ by moving $3$ inside the logarithm.

$\log_{5} ({(\frac{u v^{2}}{w^{3}})}^{3})$

Step 2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{u v^{2}}{w^{3}}$ .

$\log_{5} (\frac{{(u v^{2})}^{3}}{{(w^{3})}^{3}})$

Apply the product rule to $u v^{2}$ .

$\log_{5} (\frac{u^{3} {(v^{2})}^{3}}{{(w^{3})}^{3}})$

Step 3

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

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

$\log_{5} (\frac{u^{3} v^{2 \cdot 3}}{{(w^{3})}^{3}})$

Multiply $2$ by $3$ .

$\log_{5} (\frac{u^{3} v^{6}}{{(w^{3})}^{3}})$

Step 4

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

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

$\log_{5} (\frac{u^{3} v^{6}}{w^{3 \cdot 3}})$

Multiply $3$ by $3$ .

$\log_{5} (\frac{u^{3} v^{6}}{w^{9}})$