Answer:
x = 3
Step-by-step explanation:
Given:\(\log _{x-1}\left(16\right)=4\)
We have to solve for x.
Consider the given expression \(\log _{x-1}\left(16\right)=4\)
\(\mathrm{Apply\:log\:rule}:\quad \log _a\left(b\right)=\frac{\ln \left(b\right)}{\ln \left(a\right)}\)
\(\log _{x-1}\left(16\right)=\frac{\ln \left(16\right)}{\ln \left(x-1\right)}\)
Thus, the expression becomes,
\(\frac{\ln \left(16\right)}{\ln \left(x-1\right)}=4\)
Multiply both side by \(\ln \left(x-1\right)\)
\(\frac{\ln \left(16\right)}{\ln \left(x-1\right)}\ln \left(x-1\right)=4\ln \left(x-1\right)\)
Simplify, we get,
\(\ln \left(16\right)=4\ln \left(x-1\right)\)
Divide both side by 4, we have,
\(\frac{4\ln \left(x-1\right)}{4}=\frac{\ln \left(16\right)}{4}\)
\(\frac{\ln \left(16\right)}{4}:\quad \ln \left(2\right)\)
Thus, \(\ln \left(x-1\right)=\ln(2)\)
\(\mathrm{When\:the\:logs\:have\:the\:same\:base:\:\:}\log _b\left(f\left(x\right)\right)=\log _b\left(g\left(x\right)\right)\quad \Rightarrow \quad f\left(x\right)=g\left(x\right)\)
thus, x - 1 = 2
simplify for x, we have,
x = 3
