The required equivalent expression for [tex]{log}_{5} \ (\frac{x}{4} ) ^{2}[/tex] is [tex]2 \ log_{5}\ x-2\ log_{5}\ 4[/tex].
Given expression,
[tex]{log}_{5} \ (\frac{x}{4} ) ^{2}[/tex].
We have to find the Equivalent fraction of [tex]{log}_{5} \ (\frac{x}{4} ) ^{2}[/tex].
We know that from the properties of logarithm, the following identities
[tex]log\ m^{n} =n\ log\ m[/tex] .....(1)
[tex]log\ \frac{m}{n} = log\ m- log\ n\\[/tex].......(2)
Now, [tex]{log}_{5} \ (\frac{x}{4} ) ^{2}= 2 \ log_{5} \ \frac{x}{4}[/tex] ( using 1 identity)
Again using 2 identity, we get
[tex]{log}_{5} \ (\frac{x}{4} ) ^{2}= 2[ log_{5}\ x-log_{5}\ 4 ][/tex]. (using 2 identity)
[tex]{log}_{5} \ (\frac{x}{4} ) ^{2}= 2 \ log_{5}\ x-2\ log_{5}\ 4[/tex]
Hence the required equivalent expression for [tex]{log}_{5} \ (\frac{x}{4} ) ^{2}[/tex] is [tex]{log}_{5} \ (\frac{x}{4} ) ^{2}= 2 \ log_{5}\ x-2\ log_{5}\ 4[/tex].
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