# Question #f8e12

Dec 4, 2016

$x = 100 {e}^{-} 13.57 \approx 1.278 \times {10}^{-} 4$

#### Explanation:

We will use the following properties of logarithms and exponents:

• $\ln \left(x y\right) = \ln \left(x\right) + \ln \left(y\right)$
• ${e}^{\ln} \left(x\right) = x$
• ${e}^{-} x = \frac{1}{e} ^ x$
• ${e}^{x + y} = {e}^{x} {e}^{y}$

With those:

$\ln \left(0.01 x\right) = - 13.57$

$\implies \ln \left(0.01\right) + \ln \left(x\right) = - 13.57$

$\implies \ln \left(x\right) = - 13.57 - \ln \left(0.01\right)$

$\implies {e}^{\ln} \left(x\right) = {e}^{- 13.57 - \ln \left(0.01\right)}$

$\therefore x = {e}^{- \left(13.57 + \ln \left(0.01\right)\right)}$

$= \frac{1}{e} ^ \left(13.57 + \ln \left(0.01\right)\right)$

$= \frac{1}{{e}^{13.57} {e}^{\ln} \left(0.01\right)}$

$= \frac{1}{0.01 {e}^{13.57}}$

$= 100 {e}^{-} 13.57 \approx 1.278 \times {10}^{-} 4$