Question

Solve log base(x − 1) 16 = 4.

Answer 1

I hope this helps you


x=3

Answer 2

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