For the given equation `2^(0.1x)-5=12` , we may simplify by combining like terms.

Add `5` on both sides of the equation.

`2^(0.1x)-5+5=12+5`

`2^(0.1x)=17`

Take the "`ln` " on both sides to be able to bring down the exponent value.

Apply the natural logarithm property: `ln(x^n)= n*ln(x)` .

`ln(2^(0.1x))=ln(17)`

`0.1xln(2)=ln(17)`

`(xln(2))/10=ln(17)`

Multiply both sides by `10` .

`(xln(2))/10*10=ln(17)*10`

`xln(2)=10ln(17)`

To isolate `x` , divide both sides by `ln(2)` .

`(xln(2))/(ln(2))=(10ln(17))/(ln(2))`

`x=(10ln(17))/(ln(2)) or40.87 ` (approximated value)

Checking: Plug-in `x=40.87` on `2^(0.1x)-5=12` .

`2^(0.1*40.87)-5=?12`

`2^(4.087)-5=?12`

`17-5=?12`

`12=12 ` **TRUE**

Note: `2^(4.087)=16.99454698~~17`.

Therefore, there is no extraneous solution.

The `x=(10ln(17))/(ln(2))` is the **real exact solution** of the given equation `2^(0.1x)-5=12` .