Answer:
The Expression is given below as
[tex]\log_{\frac{1}{7}}4[/tex]
Represent the expression above to be
[tex]=x[/tex]
That is, we will have that
[tex]\log_{\frac{1}{7}}4=x[/tex]
Applying the change of base rule below, we will have that
[tex]\begin{gathered} \log_ab=y \\ b=a^y \\ lnb=lna^y \\ lnb=ylna \\ y=\frac{lnb}{lna} \end{gathered}[/tex][tex]\begin{gathered} \log_{\frac{1}{7}}4=x \\ (\frac{1}{7})^x=4 \\ (7^{-1})^x=4 \\ 7^{-x}=4 \\ take\text{ ln of both sides} \\ ln7^{-x}=ln4 \\ -xln7=ln4 \\ dividie\text{ both sides by -ln7} \\ \frac{-xln7}{-ln7}=\frac{ln4}{-ln7} \\ x=-0.712 \end{gathered}[/tex]
Hence,
The final answer is
[tex]\rightarrow-0.712[/tex]
