Answered By Leonardo F

This is a question involving the properties of logarithms. If we recall the following exponent property:

 $\log_aN^b=b\times\log_aN$log_aN^b=b×log_aN 

The exponent inside the log becomes a multiplier. Another property we have to remember is the fracional exponent and radical:

 $N^{\frac{a}{b}}=\sqrt[b]{N^a}$N^ab =^b√N^a 

Applying these two properties in this exercise:

 $8\times\log_k\left[\left(\sqrt[5]{k}\right)^p\right]=8\times\log_k\left[\left(k^{\frac{1}{5}}\right)^p\right]=8\times\log_k\left(k^{\frac{p}{5}}\right)$8×log‍_k[(⁵√k)^p]=8×log_k[(k¹⁵ )^p]=8×log_k(k^p5 ) 

Now, applying the exponent property:

 $8\times\log_k\left(k^{\frac{p}{5}}\right)=8\times\left(\frac{p}{5}\right)\times\log_kk$8×log_k(k^p5 )=8×(p5 )×log_kk 

Since  $\log_kk=1$log_kk=1 , we have the following expression:

 $\log_k\left[\left(\sqrt[5]{k}\right)^p\right]=8\times\left(\frac{p}{5}\right)=\frac{8p}{5}$log_k[(⁵√k)^p]=8×(p5 )=8p‍5  

Hence, the logarithm expression can be simplified to  $\frac{8p}{5}$8p5  .