To evaluate the equation `ln(x+19)=ln(7x-8)` , we apply natural logarithm property: `e^(ln(x))=x` .

Raise both sides by base of `e` .

`e^(ln(x+19))=e^(ln(7x-8))`

`x+19=7x-8`

Subtract `7x` from both sides of the equation.

`x+19-7x=7x-8-7x`

`-6x+19=-8`

Subtract 19 from both sides of the equation.

`-6x+19-19=-8-19`

`-6x=-27`

Divide both sides by `-6` .

`(-6x)/(-6)=(-27)/(-6)`

`x=9/2`

Checking: Plug-in `x=9/2` on `ln(x+19)=ln(7x-8)` .

`ln(9/2+19)=?ln(7*9/2-8)`

`ln(9/2+38/2)=?ln(63/2-16/2)`

`ln(47/2)=ln(47/2) ` **TRUE**

Thus, the `x=9/2` is the **real exact
solution** of the equation `ln(x+19)=ln(7x-8)` . There
is *no extraneous solution*.

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