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

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

Step 1

Simplify each term.

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

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

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

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

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

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

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

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

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

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

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

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

Step 2

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

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

Step 3

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

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Combine.

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

Step 6

Multiply $x^{\frac{2}{3}}$ by $1$ .

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