Answer:
Option (d) is correct.
The value of given expression [tex]log_{\left\{16\right\}}8[/tex] is [tex]\frac{3}{4}[/tex]
Step-by-step explanation:
Given : expression [tex]log_{\left\{16\right\}}8[/tex]
We have to evaluate the value of given expression [tex]log_{\left\{16\right\}}8[/tex]
Consider the given expression [tex]log_{\left\{16\right\}}8[/tex]
Rewrite 16 as power- base form as [tex]2^4[/tex]
Apply log rule , [tex]\log _{a^b}\left(x\right)=\frac{1}{b}\log _a\left(x\right)[/tex]
We have,
[tex]\log _{2^4}\left(8\right)=\frac{1}{4}\log _2\left(8\right)[/tex]
Simplify, we have,
[tex]=\frac{1}{4}\log _2\left(8\right)[/tex]
Rewrite 8 as power- base form as [tex]2^3[/tex]
[tex]=\frac{1}{4}\log _2\left(2^3\right)[/tex]
Apply log rule, [tex]\log _a\left(x^b\right)=b\cdot \log _a\left(x\right)[/tex]
We have,
[tex]\log _2\left(2^3\right)=3\log _2\left(2\right)[/tex]
Apply log rule, [tex]\log _a\left(a\right)=1[/tex]
We have,
[tex]\frac{1}{4}\cdot \:3\cdot \:1=\frac{3}{4}[/tex]
Thus, The value of given expression [tex]log_{\left\{16\right\}}8[/tex] is [tex]\frac{3}{4}[/tex]
