For the given equation `10^(3x)+4 =9` , we may simplify by combining like terms.

Subtract 4 from both sides of the equation.

`10^(3x)+4-4 =9-4`

`10^(3x)=5`

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(10^(3x))=ln(5)`

`3xln(10)=ln(5)`

To isolate the `x` , divide both sides by `3ln(10).`

`(3xln(10))/(3ln(10))=(ln(5))/(3ln(10))`

`x=(ln(5))/(3ln(10))`

`x= (ln(5))/(ln(1000)) or 0.233` (approximated value).

Checking: Plug-in `x=0.233` on `10^(3x)+4 =9.`

`10^(3*0.233)+4 =?9`

`10^(0.699)+4 =?9`

`5.00034535+4=?9`

`9.00034565~~9` **TRUE**.

Therefore, there is *no extraneous solution*.

The `x=(ln(5))/(3ln(10))` is the **real exact solution** of the given equation `10^(3x)+4 =9` .

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