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