Question

Solve log base(x − 1) 16 = 4.

Answer 1

i hope this helps you


x=3

Answer 2

x = 3

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)=(\ln \left(b\right))/(\ln \left(a\right))`

`\log _(x-1)\left(16\right)=(\ln \left(16\right))/(\ln \left(x-1\right))`

Thus, the expression becomes,

`(\ln \left(16\right))/(\ln \left(x-1\right))=4`

Multiply both side by
`\ln \left(x-1\right)`

`(\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,

`(4\ln \left(x-1\right))/(4)=(\ln \left(16\right))/(4)`

`(\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