Question

Precalc.

Solve the equation for x, accurate to three decimal places: 4^2 − x = 7^3 − x

multi-step equations

Answer 1

`4^(2-x)` and
`7^(3-x)` are positive for all x∈R, so we can logarithm both sides. There will be:


`4^(2-x)=7^(3-x)\quad|\ln(\dots)\\\\\ln4^(2-x)=\ln7^(3-x)\\\\(2-x)\ln4=(3-x)\ln7\\\\2\ln4-x\ln4=3\ln7-x\ln7\\\\x\ln7-x\ln4=3\ln7-2\ln4\\\\x(\ln7-\ln4)=3\ln7-2\ln4\quad|:(\ln7-\ln4)\\\\\\\boxed{x=(3\ln7-2\ln4)/(\ln7-\ln4)}`

This is exact form of solution. Now, we can use tables of logarithms or calculator and calculate approximate form. We get:


`\boxed{x\approx5,477}`