# Chapter 4 - Section 4.4 - Laws of Logarithms - 4.4 exercises: 56

$\log_5 (\frac{x^2y^4}{z^6})$

#### Work Step by Step

$Combine$ $the$ $expression$: $2$$($$\log_5 x$ + $2$$\log_5 y - 3$$\log_5 z$$) Distribute the 2 to all variables in the parenthesis 2$$\log_5 x$ + $2$$\times$$2$$\log_5 y - 2$$\times$$3$$\log_5 z$ $2$$\log_5 x + 4$$\log_5 y$ - $6$$\log_5 z Apply the Third Law of Logarithms for 2$$\log_5 x$, $4$$\log_5 y, and 6$$\log_5 z$ $2$$\log_5 x = \log_5 x^2 4$$\log_5 y$ = $\log_5 y^4$ $6$$\log_5 z$ = $\log_5 x^6$ $\log_5 x^2$ + $\log_5 y^4$ - $\log_5 z^6$ Apply the First Law of Logarithms for $\log_5 x^2$ + $\log_5 y^4$ $\log_5 x^2$ + $\log_5 y^4$ = $\log_5 (x^2\times y^4)$ $\log_5 (x^2y^4)$ - $\log_5 z^6$ Apply the Second Law of Logarithms $\log_5 (x^2y^4)$ - $\log_5 z^6$ = $\log_5 (\frac{x^2y^4}{z^6})$

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