# What is x if log(7x-12) - 2 log(x)= 1 ?

Oct 22, 2015

Imaginary Roots

#### Explanation:

I think roots are imaginary
You may know that $\log {a}^{n} = n \log a$
So, $2 \log x = \log {x}^{2}$
Thus the equation becomes
$\log \left(7 x - 12\right) - \log {x}^{2} = 1$
Also you may know
$\log a - \log c = \log \left(\frac{a}{c}\right)$
Hence the equation reduces to
log $\frac{7 x - 12}{x} ^ 2 = 1$
You may also know,
if log a to base b is = c, then
$a = {b}^{c}$
For $\log x$ the base is 10
So the equation reduces to
$\frac{7 x - 12}{x} ^ 2 = {10}^{1} = 10$
or
$\left(7 x - 12\right) = 10 \cdot {x}^{2}$
ie $10 \cdot {x}^{2} - 7 x + 12 = 0$
This is a quadratic equation and the roots are imaginary, since $4 \cdot 10 \cdot 12 > {7}^{2}$