Question

Which expression is equivalent to log Subscript 5 Baseline (StartFraction x Over 4 EndFraction) squared? 2 log Subscript 5 Baseline x log Subscript 5 Baseline 4 2 log Subscript 5 Baseline x log Subscript 5 Baseline 16 2 log Subscript 5 Baseline x 2 log Subscript 5 Baseline 4 2 log Subscript 5 Baseline x log Subscript 5 Baseline 4.

Answer 1

The required equivalent expression for  ${log}_{5} \ (\frac{x}{4} ) ^{2}$ is $2 \ log_{5}\ x-2\ log_{5}\ 4$.

Given expression,

${log}_{5} \ (\frac{x}{4} ) ^{2}$.

We have to find the Equivalent fraction of  ${log}_{5} \ (\frac{x}{4} ) ^{2}$.

We know that from the properties of logarithm, the following identities

$log\ m^{n} =n\ log\ m$ .....(1)

$log\ \frac{m}{n} = log\ m- log\ n$.......(2)

Now, ${log}_{5} \ (\frac{x}{4} ) ^{2}= 2 \ log_{5} \ \frac{x}{4}$ ( using 1 identity)

Again using 2 identity, we get

${log}_{5} \ (\frac{x}{4} ) ^{2}= 2[ log_{5}\ x-log_{5}\ 4 ]$. (using 2 identity)

${log}_{5} \ (\frac{x}{4} ) ^{2}= 2 \ log_{5}\ x-2\ log_{5}\ 4$

Hence the required equivalent expression for  ${log}_{5} \ (\frac{x}{4} ) ^{2}$ is ${log}_{5} \ (\frac{x}{4} ) ^{2}= 2 \ log_{5}\ x-2\ log_{5}\ 4$.

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