which expression is equivalent to $log_{5}(\frac{x}{4})^{2}$?\n$2log_{5}x + log_{5}4$\n$2log_{5}x+log_{5}16$\…

which expression is equivalent to $log_{5}(\frac{x}{4})^{2}$?\n$2log_{5}x + log_{5}4$\n$2log_{5}x+log_{5}16$\n$2log_{5}x - 2log_{5}4$\n$2log_{5}x-log_{5}4$

which expression is equivalent to $log_{5}(\frac{x}{4})^{2}$?\n$2log_{5}x + log_{5}4$\n$2log_{5}x+log_{5}16$\n$2log_{5}x - 2log_{5}4$\n$2log_{5}x-log_{5}4$

Answer

Explanation:

Step1: Apply power - rule of logarithms

According to the power - rule $\log_aM^n=n\log_aM$, for $\log_5(\frac{x}{4})^2$, we have $2\log_5\frac{x}{4}$.

Step2: Apply quotient - rule of logarithms

The quotient - rule is $\log_a\frac{M}{N}=\log_aM-\log_aN$. So, $2\log_5\frac{x}{4}=2(\log_5x - \log_54)$.

Step3: Distribute the 2

Using the distributive property $a(b - c)=ab - ac$, we get $2\log_5x-2\log_54$.

Answer:

$2\log_5x - 2\log_54$ (the third option)