Simplify/Condense 4/3 log base 5 of x-5/4 log base 5 of y-3/2 log base 5 of z

$\frac{4}{3} \log_{5} (x) - \frac{5}{4} \log_{5} (y) - \frac{3}{2} \log_{5} (z)$

Simplify each term.

Simplify $\frac{4}{3} \log_{5} (x)$ by moving $\frac{4}{3}$ inside the logarithm.

$\log_{5} (x^{\frac{4}{3}}) - \frac{5}{4} \cdot \log_{5} (y) - \frac{3}{2} \cdot \log_{5} (z)$

Multiply $- \frac{5}{4} \log_{5} (y)$ .

Reorder $\log_{5} (y)$ and $\frac{5}{4}$ .

$\log_{5} (x^{\frac{4}{3}}) - (\frac{5}{4} \log_{5} (y)) - \frac{3}{2} \cdot \log_{5} (z)$

Simplify $\frac{5}{4} \log_{5} (y)$ by moving $\frac{5}{4}$ inside the logarithm.

$\log_{5} (x^{\frac{4}{3}}) - \log_{5} (y^{\frac{5}{4}}) - \frac{3}{2} \cdot \log_{5} (z)$

Multiply $- \frac{3}{2} \log_{5} (z)$ .

Reorder $\log_{5} (z)$ and $\frac{3}{2}$ .

$\log_{5} (x^{\frac{4}{3}}) - \log_{5} (y^{\frac{5}{4}}) - (\frac{3}{2} \log_{5} (z))$

Simplify $\frac{3}{2} \log_{5} (z)$ by moving $\frac{3}{2}$ inside the logarithm.

$\log_{5} (x^{\frac{4}{3}}) - \log_{5} (y^{\frac{5}{4}}) - \log_{5} (z^{\frac{3}{2}})$

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

$\log_{5} (\frac{x^{\frac{4}{3}}}{y^{\frac{5}{4}}}) - \log_{5} (z^{\frac{3}{2}})$

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

$\log_{5} (\frac{\frac{x^{\frac{4}{3}}}{y^{\frac{5}{4}}}}{z^{\frac{3}{2}}})$

Multiply the numerator by the reciprocal of the denominator.

$\log_{5} (\frac{x^{\frac{4}{3}}}{y^{\frac{5}{4}}} \cdot \frac{1}{z^{\frac{3}{2}}})$

Combine.

$\log_{5} (\frac{x^{\frac{4}{3}} \cdot 1}{y^{\frac{5}{4}} z^{\frac{3}{2}}})$

Multiply $x^{\frac{4}{3}}$ by $1$ .

$\log_{5} (\frac{x^{\frac{4}{3}}}{y^{\frac{5}{4}} z^{\frac{3}{2}}})$