Write as a Single Logarithm 5 log base b of x+9 log base b of y-3 log base b of z

$5 \log_{b} (x) + 9 \log_{b} (y) - 3 \log_{b} (z)$

Step 1

Simplify each term.

Simplify $5 \log_{b} (x)$ by moving $5$ inside the logarithm.

$\log_{b} (x^{5}) + 9 \log_{b} (y) - 3 \log_{b} (z)$

Simplify $9 \log_{b} (y)$ by moving $9$ inside the logarithm.

$\log_{b} (x^{5}) + \log_{b} (y^{9}) - 3 \log_{b} (z)$

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

$\log_{b} (x^{5}) + \log_{b} (y^{9}) - \log_{b} (z^{3})$

Step 2

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

$\log_{b} (x^{5} y^{9}) - \log_{b} (z^{3})$

Step 3

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

$\log_{b} (\frac{x^{5} y^{9}}{z^{3}})$