Write as a Single Logarithm 2 log base b of u+3 log base b of v-1/2 log base b of w

$2 \log_{b} (u) + 3 \log_{b} (v) - \frac{1}{2} \log_{b} (w)$

Step 1

Simplify each term.

Simplify $2 \log_{b} (u)$ by moving $2$ inside the logarithm.

$\log_{b} (u^{2}) + 3 \log_{b} (v) - \frac{1}{2} \cdot \log_{b} (w)$

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

$\log_{b} (u^{2}) + \log_{b} (v^{3}) - \frac{1}{2} \cdot \log_{b} (w)$

Multiply $- \frac{1}{2} \log_{b} (w)$ .

Reorder $\log_{b} (w)$ and $\frac{1}{2}$ .

$\log_{b} (u^{2}) + \log_{b} (v^{3}) - (\frac{1}{2} \log_{b} (w))$

Simplify $\frac{1}{2} \log_{b} (w)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log_{b} (u^{2}) + \log_{b} (v^{3}) - \log_{b} (w^{\frac{1}{2}})$

Step 2

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

$\log_{b} (u^{2} v^{3}) - \log_{b} (w^{\frac{1}{2}})$

Step 3

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

$\log_{b} (\frac{u^{2} v^{3}}{w^{\frac{1}{2}}})$