# Question #0c8b4

Dec 8, 2014

The application of the properties of logarithms is needed here to solve for the value of x.

$\log a - \log b = \log \left(\frac{a}{b}\right)$
This is a common logarithm. Therefore, the base is 10

Applying this property,

$\log 3 \left(x\right) - \log 3 \left(x + 6\right) = - 1$

$\log \left[\frac{3 x}{3 \left(x + 6\right)}\right] = - 1$

then applying log b = y, which is ${10}^{y} = b$

Therefore,

$\left[\frac{3 x}{3 \left(x + 6\right)}\right] = {10}^{-} 1$

take note,${a}^{-} 1 = \left(\frac{1}{a}\right)$

Simplifying,
$\frac{3 x}{3 x + 18} = \frac{1}{10}$

then we cross multiply

$10 \cdot 3 x = 1 \cdot \left(3 x + 18\right)$
$30 x = 3 x + 18$

add -3x to both sides

$30 x - 3 x = 3 x + \left(- 3 x\right) + 18$

then,

$27 x = 18$

divide by 27

$x = \left(\frac{18}{27}\right)$

reducing to lowest term

$x = \frac{2}{3}$

we are asked to round off to two decimals. Therefore the answer is $0.67$