# How do you solve ln sqrt(x-8)= 5?

Jun 21, 2016

We go step by step un-doing the things that are being done to the variable, $x$, making sure that we do the same thing to both sides arriving at the answer:

$x = {e}^{10} + 8$

#### Explanation:

We go step by step un-doing the things that are being done to the variable, $x$, making sure that we do the same thing to both sides. The first thing we encounter is the natural logarithm, which we can un-do using it's inverse, ${e}^{x}$. Starting with the left hand side:

${e}^{\ln} \left(\sqrt{x - 8}\right) = \sqrt{x - 8}$

then the right hand side is:

${e}^{5}$

Which makes our equation:

$\sqrt{x - 8} = {e}^{5}$

Now we un-do the square-root by squaring both sides. Starting with the left hand side:

${\left(\sqrt{x - 8}\right)}^{2} = x - 8$

and the right hand side:

${\left({e}^{5}\right)}^{2} = {e}^{5 \cdot 2} = {e}^{10}$

which makes our equation:

$x - 8 = {e}^{10}$

Now we can add $8$ to both sides giving:

$x = {e}^{10} + 8$