Question

How do you solve `6^(x-2)=4^x`?

Answer 1

`x=8.838`

Explanation:

In the equation `6^(x-2)=4^x`, taking log on both sides we get

`(x-2)log6=xlog4`

or `xlog6-2log6=xlog4`

or `xlog6-xlog4=2log6`

or `x=(2log6)/(log6-log4)`

= `(2xx0.7782)/(0.7782-0.6021)`

= `1.5564/0.1761=8.838`

Answer 2

`x=8.838`

Explanation:

`6^(x-2)=4^x`

Log both sides.

`log 6^(x-2)=log 4^x`

Recall the log rule `logx^a = alogx`

`(x-2) log6 = xlog4`

Distribute the `log6`

`xlog6-2log6=xlog4`

Subtract `xlog6` from both sides.

`-2log6 = xlog4 - xlog6`

Factor out the `x` on the right side

-2log6=x(log4-log6)

Recall the log rule `loga - logb= log (a/b)`

`-2log6=xlog(4/6)`

`-2log6=xlog(2/3)`

Divide both sides by `log (2/3)`

`frac{-2log6}{log (2/3)}=x`

Use a calculator....

`x=8.838`