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`2^(0.1x)-5=12` Solve the equation.

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` .

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