Write as a Single Logarithm 3 log base b of 4x-( log base b of 5y+ log base b of 6z)

$3 \log_{b} (4 x) - (\log_{b} (5 y) + \log_{b} (6 z))$

Simplify each term.

Simplify $3 \log_{b} (4 x)$ by moving $3$ inside the logarithm.

$\log_{b} ({(4 x)}^{3}) - (\log_{b} (5 y) + \log_{b} (6 z))$

Apply the product rule to $4 x$ .

$\log_{b} (4^{3} x^{3}) - (\log_{b} (5 y) + \log_{b} (6 z))$

Raise $4$ to the power of $3$ .

$\log_{b} (64 x^{3}) - (\log_{b} (5 y) + \log_{b} (6 z))$

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

$\log_{b} (64 x^{3}) - \log_{b} (5 y (6 z))$

Rewrite using the commutative property of multiplication.

$\log_{b} (64 x^{3}) - \log_{b} (5 \cdot 6 y z)$

Multiply $5$ by $6$ .

$\log_{b} (64 x^{3}) - \log_{b} (30 y z)$

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

$\log_{b} (\frac{64 x^{3}}{30 y z})$

Cancel the common factor of $64$ and $30$ .

Factor $2$ out of $64 x^{3}$ .

$\log_{b} (\frac{2 (32 x^{3})}{30 y z})$

Cancel the common factors.

Factor $2$ out of $30 y z$ .

$\log_{b} (\frac{2 (32 x^{3})}{2 (15 y z)})$

Cancel the common factor.

$\log_{b} (\frac{2 (32 x^{3})}{2 (15 y z)})$

Rewrite the expression.

$\log_{b} (\frac{32 x^{3}}{15 y z})$